# The Ultimate Mathematics Cheat Sheet Collection

## Algebraic Expressions

### Basic Algebraic Expressions

• Addition: To add two or more terms, just combine the coefficients of the like terms. For example, 2x + 3x = 5x
• Subtraction: To subtract two or more terms, just subtract the coefficients of the like terms. For example, 2x - 3x = -x
• Multiplication: To multiply two or more terms, just multiply the coefficients and add the exponents of the like variables. For example, 2x * 3x = 6x^2
• Division: To divide two or more terms, just divide the coefficients and subtract the exponents of the like variables. For example, (6x^2) / (2x) = 3x

### Common Algebraic Expressions

• Distributive Property: a(b+c) = ab + ac
• Factoring: To factor a polynomial, find the common factor of all the terms and divide each term by that factor. For example, 6x^2 + 9x = 3x(2x + 3)
• FOIL Method: (a + b)(c + d) = ac + ad + bc + bd
• Quadratic Formula: The quadratic formula can be used to find the roots of a quadratic equation. The formula is x = (-b ± sqrt(b^2 - 4ac)) / 2a

### Rules of Exponents

• Product Rule: a^m * a^n = a^(m+n)
• Power Rule: (a^m)^n = a^(mn)
• Quotient Rule: a^m / a^n = a^(m-n)
• Negative Exponents: a^-m = 1 / a^m

## Algebraic Equations Cheat Sheet

### Linear Equations

A linear equation is an equation of the form Ax + By = C, where A, B, and C are constants, and x and y are variables.

• Solving Linear Equations: To solve a linear equation for one variable, use inverse operations to isolate the variable on one side of the equation. For example, to solve 2x + 3 = 7, subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to get x = 2.
• Slope-Intercept Form: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
• Point-Slope Form: The point-slope form of a linear equation is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.

• Quadratic Formula: The quadratic formula can be used to find the roots of a quadratic equation. The formula is x = (-b ± sqrt(b^2 - 4ac)) / 2a
• Completing the Square: Completing the square is a method for solving quadratic equations. To complete the square, add and subtract (b/2a)^2 to the equation, then factor the resulting trinomial. For example, to solve x^2 + 6x + 5 = 0, add and subtract (6/2)^2 = 9 to get (x + 3)^2 - 4 = 0, then factor to get (x + 3 - 2)(x + 3 + 2) = 0, which gives x = -5 or x = -1.

### Systems of Equations

A system of equations is a set of equations with multiple variables.

• Substitution Method: To solve a system of equations using the substitution method, solve one of the equations for one of the variables, then substitute the expression for that variable into the other equation. For example, to solve the system x + y = 7 and 2x - y = 1, solve the first equation for y to get y = 7 - x, then substitute 7 - x for y in the second equation to get 2x - (7 - x) = 1. Solve for x to get x = 2, then substitute x = 2 into either equation to get y = 5.
• Elimination Method: To solve a system of equations using the elimination method, add or subtract the equations to eliminate one of the variables, then solve for the remaining variable. For example, to solve the system 3x + 2y = 10 and 2x - 4y = -4, multiply the first equation by 2 to get 6x + 4y = 20, then add the second equation to eliminate y and get 8x = 16. Solve for x to get x = 2, then substitute x = 2 into either equation to get y = 2.

### Exponential Equations

An exponential equation is an equation with a variable in the exponent.

• Solving Exponential Equations: To solve an exponential equation, take the logarithm of both sides of the equation. For example, to solve 2^x = 16, take the logarithm base 2 of both sides to get x = 4.
• Properties of Exponents: The properties of exponents include the product rule (a^m * a^n = a^(m+n)), the quotient rule (a^m / a^n = a^(m-n)), and the power rule ((a^m)^n = a^(mn)).

## Linear Equations

A linear equation is an equation of the form Ax + By = C, where A, B, and C are constants, and x and y are variables.

### Solving Linear Equations

To solve a linear equation for one variable, use inverse operations to isolate the variable on one side of the equation. For example, to solve 2x + 3 = 7, subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to get x = 2.

### Slope-Intercept Form

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

• Finding the Slope: The slope of a line is the ratio of the change in y to the change in x, or rise over run. It can also be found by comparing the coefficients of x and y in the equation.
• Finding the Y-Intercept: The y-intercept is the point where the line intersects the y-axis. It can be found by setting x = 0 in the equation and solving for y.
• Graphing a Line: To graph a line in slope-intercept form, plot the y-intercept, then use the slope to find other points on the line.

### Point-Slope Form

The point-slope form of a linear equation is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

• Finding the Slope: The slope can be found using the coordinates of the two points on the line.
• Finding the Equation: Substitute the slope and coordinates of a point on the line into the point-slope form and simplify to get the equation in slope-intercept form.

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is a variable.

There are several methods for solving quadratic equations, including:

• Factoring: If a quadratic equation can be factored, set each factor equal to zero and solve for x. For example, to solve x^2 + 5x + 6 = 0, factor as (x + 2)(x + 3) = 0, then set each factor equal to zero and solve to get x = -2 or x = -3.
• Completing the Square: To solve a quadratic equation by completing the square, rewrite the equation in the form a(x - h)^2 + k = 0, where (h, k) is the vertex of the parabola. Solve for x by taking the square root of both sides and adding or subtracting h as needed. For example, to solve x^2 + 6x - 8 = 0, complete the square to get (x + 3)^2 - 17 = 0, then solve to get x = -3 + sqrt(17) or x = -3 - sqrt(17).
• Quadratic Formula: The quadratic formula can be used to solve any quadratic equation of the form ax^2 + bx + c = 0. The formula is x = (-b ± sqrt(b^2 - 4ac)) / 2a. For example, to solve 2x^2 + 3x - 2 = 0, use the quadratic formula to get x = -3/4 ± sqrt(17)/4.

### Vertex Form

The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola and a is a scaling factor that determines the width and direction of the parabola.

• Finding the Vertex: The vertex can be found by completing the square or by using the formula h = -b/2a and k = f(h).
• Finding the Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. It can be found by using the formula x = -b/2a.
• Graphing a Parabola: To graph a parabola in vertex form, plot the vertex and use the scaling factor to find other points on the parabola.

## Exponential Functions

An exponential function is a function of the form f(x) = a^x, where a is a constant and x is a variable.

### Properties of Exponential Functions

• Domain: The domain of an exponential function is all real numbers.
• Range: The range of an exponential function depends on the value of a. If a > 0, then the range is (0, infinity). If a < 0, then the range is (-infinity, 0).
• Increasing/Decreasing: If a > 1, then the function is increasing. If 0 < a < 1, then the function is decreasing.
• Asymptotes: An exponential function has a horizontal asymptote at y = 0 if 0 < a < 1, and no horizontal asymptote if a > 1.

### Graphing Exponential Functions

To graph an exponential function, you can use a table of values or plot a few points and sketch the graph.

• Table of Values: To create a table of values, choose some x-values and calculate the corresponding y-values using the formula f(x) = a^x. Plot the points and sketch the graph.
• Points: To plot a few points, choose some x-values and calculate the corresponding y-values using the formula f(x) = a^x. Plot the points and sketch the graph, making sure to include the asymptote if necessary.

### Exponential Growth and Decay

Exponential functions can be used to model exponential growth and decay.

• Growth: If a > 1, then the function represents exponential growth. The formula for exponential growth is f(x) = ab^x, where a is the initial amount and b is the growth factor.
• Decay: If 0 < a < 1, then the function represents exponential decay. The formula for exponential decay is f(x) = ab^x, where a is the initial amount and b is the decay factor.
• Half-Life: The half-life of an exponential function is the time it takes for the function to decay to half its initial value. The formula for half-life is t = (ln 2) / k, where t is the half-life, ln is the natural logarithm, and k is the decay constant.

## Logarithmic Functions Cheat Sheet

A logarithmic function is the inverse of an exponential function. It is a function of the form f(x) = log_a(x), where a is a constant and x is a variable.

### Properties of Logarithmic Functions

Property Formula
Domain: x > 0
Range: all real numbers
Increasing/Decreasing: If 0 < a < 1, then the function is decreasing. If a > 1, then the function is increasing.
Asymptotes: A logarithmic function has a vertical asymptote at x = 0 and no horizontal asymptote.

### Graphing Logarithmic Functions

To graph a logarithmic function, you can use a table of values or plot a few points and sketch the graph.

• Table of Values: To create a table of values, choose some x-values and calculate the corresponding y-values using the formula f(x) = log_a(x). Plot the points and sketch the graph.
• Points: To plot a few points, choose some x-values and calculate the corresponding y-values using the formula f(x) = log_a(x). Plot the points and sketch the graph, making sure to include the asymptote.

### Logarithmic Identities

Logarithmic identities are useful when simplifying or solving equations involving logarithmic functions.

Identity Formula
Product Rule: log_a(xy) = log_a(x) + log_a(y)
Quotient Rule: log_a(x/y) = log_a(x) - log_a(y)
Power Rule: log_a(x^k) = k*log_a(x)
Change of Base: log_a(x) = (log_b(x))/(log_b(a))

## Trigonometric Functions Cheat Sheet

Trigonometric functions are used to model periodic phenomena such as sound and light waves, as well as in navigation and engineering. The six basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.

### Definitions of Trigonometric Functions

For an angle θ in a right triangle with hypotenuse of length 1, the six trigonometric functions are defined as follows:

Function Definition
Sine: sin(θ) = opposite/hypotenuse
Cosecant: csc(θ) = 1/sin(θ) = hypotenuse/opposite
Secant: sec(θ) = 1/cos(θ) = hypotenuse/adjacent
Cotangent: cot(θ) = 1/tan(θ) = adjacent/opposite

### Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is useful for understanding trigonometric functions because the coordinates of the point on the circle intersected by an angle θ are (cos(θ), sin(θ)).

For example, the angle θ = π/6 intersects the unit circle at the point (sqrt(3)/2, 1/2), so sin(π/6) = 1/2 and cos(π/6) = sqrt(3)/2.

### Trigonometric Identities

Trigonometric identities are used to simplify or solve equations involving trigonometric functions.

Identity Formula
Pythagorean: sin^2(θ) + cos^2(θ) = 1
Reciprocal: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ)
Quotient: tan(θ) = sin(θ)/cos(θ)
Even/Odd: cos(-θ) = cos(θ), sin(-θ) = -sin(θ)
Double Angle: sin(2θ) = 2sin(θ)cos(θ), cos(2θ) = cos^2(θ) - sin^2(θ)
Half Angle: sin(θ/2) = ±sqrt((1-cos(θ))/2), cos(θ/2) = ±sqrt((1+cos(θ))/2)

## Trigonometric Functions of Common Angles

Trigonometric functions of common angles are useful for quick calculations.

Angle Sine Cosine Tangent
0 0 1 0
π/6 1/2 sqrt(3)/2 sqrt(3)/3
π/4 sqrt(2)/2 sqrt(2)/2 1
π/3 sqrt(3)/2 1/2 sqrt(3)
π/2 1 0 undefined

## Limits

A limit is a value that a function approaches as the input variable gets closer to a certain value. It is an important concept in calculus and is used to define derivatives and integrals.

### Limit Notation

The notation used to represent a limit is:

limx→a f(x) = L

which means that as x approaches a, the value of f(x) approaches L.

### Limit Properties

There are several properties of limits that are useful in calculus:

Property Example Explanation
Sum/Difference limx→a (f(x) ± g(x)) = limx→a f(x) ± limx→a g(x) The limit of a sum or difference is the sum or difference of the limits.
Product limx→a (f(x)g(x)) = (limx→a f(x))(limx→a g(x)) The limit of a product is the product of the limits.
Quotient limx→a (f(x)/g(x)) = (limx→a f(x))/(limx→a g(x)) The limit of a quotient is the quotient of the limits, provided that the limit of the denominator is not zero.
Power limx→a (f(x)^n) = (limx→a f(x))^n The limit of a power is the power of the limit.
Root limx→a sqrt(f(x)) = sqrt(limx→a f(x)) The limit of a square root is the square root of the limit.
Chain Rule limx→a f(g(x)) = f(limx→a g(x)) The limit of a composite function is the composition of the limits, provided that the limit of the inner function exists at the limit point.

### Limit Laws

There are several laws of limits that are useful in calculus:

Law Example Explanation
Constant limx→a c = c The limit of a constant is the constant.
Identity limx→a x = a The limit of the identity function is the limit point.
Zero limx→a 0 = 0 The limit of zero is zero.
One limx→a 1 = 1 The limit of one is one.
Inverse limx→a 1/f(x) = 1/limx→a f(x) The limit of the inverse of a function is the inverse of the limit of the function, provided that the limit of the function is not zero.
Reciprocal limx→a f(x)/g(x) = limx→a f(x) / limx→a g(x) The limit of a ratio is the ratio of the limits, provided that the limit of the denominator is not zero.
Squeeze If g(x) ≤ f(x) ≤ h(x) for all x near a (except possibly at a) and limx→a g(x) = limx→a h(x) = L, then limx→a f(x) = L. The limit of a function squeezed between two other functions with the same limit is the same as the limit of those functions.

### Types of Limits

There are several types of limits that are useful in calculus:

• Left-hand limit: limx→a- f(x)
• Right-hand limit: limx→a+ f(x)
• Infinite limit: limx→a f(x) = ∞ or -∞
• Limit at infinity: limx→∞ f(x) or limx→-∞ f(x)

### Common Limits

There are several common limits that are useful in calculus:

Function Limit
sin(x)/x limx→0 sin(x)/x = 1
(ex - 1)/x limx→0 (ex - 1)/x = 1
ln(x) limx→0 ln(x) = -∞
1/x limx→0 1/x = ∞ or -∞
xn limx→a xn = an
√x limx→0 √x = 0
sin(x) limx→0 sin(x) = 0
cos(x) limx→0 cos(x) = 1
tan(x) limx→0 tan(x) = 0

### L'Hôpital's Rule

L'Hôpital's rule is a method of finding the limit of a function when the limit of the function and its derivative are both zero or both infinity. The rule states that:

limx→a f(x)/g(x) = limx→a f'(x)/g'(x), provided that the limit of the quotient of the derivatives exists.

In other words, to use L'Hôpital's rule:

1. Take the derivative of the numerator and the denominator.
2. Find the limit of the new fraction.
3. If the limit exists, it is the same as the limit of the original fraction.
4. If the limit does not exist, the original limit does not exist.

### Common Limits using L'Hôpital's Rule

There are several common limits that can be evaluated using L'Hôpital's rule:

Function Limit
0/0 or ∞/∞ limx→a f(x)/g(x) = limx→a f'(x)/g'(x)
∞ - ∞ or 0 * 0
ex/xn limx→∞ ex/xn = ∞
ln(x)/xn limx→0+ ln(x)/xn = 0
xn/ex limx→∞ xn/ex = 0
x * ln(x) limx→0+ x * ln(x) = 0
xn * ex limx→∞ xn * ex = ∞

## Derivatives Cheat Sheet

Function Derivative
c 0
xn n*xn-1
ex ex
ln(x) 1/x
sin(x) cos(x)
cos(x) -sin(x)
tan(x) sec2(x)
sec(x) sec(x)*tan(x)
csc(x) -csc(x)*cot(x)
cot(x) -csc2(x)
f(x) + g(x) f'(x) + g'(x)
f(x) * g(x) f'(x) * g(x) + f(x) * g'(x)
f(g(x)) f'(g(x)) * g'(x)
f(x) / g(x) (f'(x) * g(x) - f(x) * g'(x)) / g2(x)

## Integrals

Function Integral
c c*x
xn xn+1 / (n+1)
ex ex
1/x ln|x| + c
sin(x) -cos(x)
cos(x) sin(x)
tan(x) -ln|cos(x)| + c
sec(x) ln|sec(x) + tan(x)| + c
csc(x) -ln|csc(x) + cot(x)| + c
cot(x) ln|sin(x)| + c
f(x) + g(x) ∫f(x)dx + ∫g(x)dx
k * f(x) k * ∫f(x)dx
∫f(g(x))g'(x)dx ∫f(u)du
u-substitution: ∫f(g(x))g'(x)dx ∫f(u)du = ∫f(g(x))g'(x) * dx
Integration by Parts: ∫u(x)v'(x)dx u(x) * v(x) - ∫v(x)u'(x)dx

## Differential Equations

Equation Solution
y' = f(x) y = ∫f(x)dx + C
y' = k*y y = Cekx
y'' + ay' + by = 0 y = C1er1x + C2er2x
y'' + py' + qy = 0 y = e-p/2x(C1cos(w*x) + C2sin(w*x))
y' + ky = f(x) y = (C + ∫f(x)e-kxdx) * ekx
y' + p(x)y = q(x) y = e-∫p(x)dx * (∫q(x)e∫p(x)dxdx + C)
yn = f(x) y = ∫f(x)dx + C
yn + py = f(x) y = (C + ∫f(x)x-p/ndx) * xp/n
yn + py = g(x)ym y = [C + (1-m) * ∫g(x)x-p/ndx] * xp/n

## Probability

### Basic Probability Concepts

Concept Formula
Sample Space The set of all possible outcomes of an experiment
Event A subset of the sample space
Probability P(event) = (number of favorable outcomes) / (total number of outcomes)
Complement P(not A) = 1 - P(A)
Addition Rule P(A or B) = P(A) + P(B) - P(A and B)
Multiplication Rule P(A and B) = P(A) * P(B|A)
Conditional Probability P(A|B) = P(A and B) / P(B)
Independent Events P(A and B) = P(A) * P(B)

## Discrete Probability Distributions

Distribution Probability Mass Function Mean Variance
Uniform Distribution f(x) = 1 / n (a + b) / 2 (n2 - 1) / 12
Binomial Distribution f(x) = (nCx) * px * (1-p)n-x np np(1-p)
Poisson Distribution f(x) = (e * λx) / x! λ λ
Geometric Distribution f(x) = p * (1-p)x-1 1/p (1-p) / p2
Conditional Probability P(A|B) = P(A ∩ B) / P(B)
Bayes' Theorem P(A|B) = P(B|A) * P(A) / P(B)
Law of Total Probability P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn)
Expected Value E(X) = Σi=1n xi * P(X=xi)
Variance Var(X) = E(X2) - [E(X)]2
Standard Deviation SD(X) = √(Var(X))

This cheat sheet covers the basics of probability theory, including definitions of terms like random variable, probability function, and expectation. It also includes formulas for computing probabilities, such as the addition and multiplication rules, as well as conditional probability and Bayes' theorem. Additionally, it includes formulas for calculating expected value, variance, and standard deviation. This cheat sheet serves as a useful reference for students and professionals working in fields where probability theory is applicable, such as statistics, economics, and finance.

## Statistics Cheat Sheet

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. It involves gathering and summarizing numerical information in order to make informed decisions. This cheat sheet provides an overview of some of the key concepts and formulas in statistics.

Concept Formula
Population Mean μ = (Σx) / N
Sample Mean x̄ = (Σx) / n
Population Variance σ2 = (Σ(x - μ)2) / N
Sample Variance s2 = (Σ(x - x̄)2) / (n - 1)
Standard Deviation σ = √σ2
Sample Standard Deviation s = √s2
Normal Distribution P(a < X < b) = Φ((b - μ) / σ) - Φ((a - μ) / σ)
t-Distribution t = (x̄ - μ) / (s / √n)
Confidence Interval x̄ ± tα/2 * (s / √n)
Margin of Error tα/2 * (s / √n)
Hypothesis Testing t = (x̄ - μ0) / (s / √n)
p-Value p = P(|T| ≥ |t|)
Type I Error Rejecting a true null hypothesis (false positive)
Type II Error Accepting a false null hypothesis (false negative)

This cheat sheet covers some of the key concepts and formulas in statistics, including the mean, variance, and standard deviation for populations and samples. It also includes formulas for working with normal and t-d

## Permutations and Combinations Cheat Sheet

Permutations and combinations are important concepts in mathematics that deal with the arrangement and selection of objects. Permutations refer to the arrangement of objects in a specific order, whereas combinations refer to the selection of objects without considering their order. Here is a cheat sheet that provides the formulas and definitions for permutations and combinations.

### Permutations:

Permutations refer to the number of ways in which a set of objects can be arranged in a specific order. The formula for permutations is:

nPr = n! / (n - r)!

where n is the total number of objects and r is the number of objects to be arranged.

### Combinations:

Combinations refer to the number of ways in which a set of objects can be selected without considering their order. The formula for combinations is:

nCr = n! / (r! * (n - r)!)

where n is the total number of objects and r is the number of objects to be selected.

### Permutations vs Combinations:

Permutations and combinations are different in terms of their order. Permutations take into account the order of the objects, while combinations do not. For example, consider the set {A, B, C}. The permutations of this set would be:

ABC, ACB, BAC, BCA, CAB, CBA

The combinations of this set would be:

AB, AC, BC

Notice that the order of the objects does not matter in combinations.

### Permutations with repetition:

When the objects in a set can be repeated, the formula for permutations with repetition is:

nr

where n is the number of objects and r is the number of times they can be repeated.

### Combinations with repetition:

When the objects in a set can be repeated, the formula for combinations with repetition is:

(n + r - 1)Cr

where n is the number of objects and r is the number of times they can be repeated.

## Set Theory

### Set Theory Operations

• Union: The union of two sets A and B is the set of all elements that belong to A or B or both. It is denoted by A ∪ B.
• Intersection: The intersection of two sets A and B is the set of all elements that belong to both A and B. It is denoted by A ∩ B.
• Complement: The complement of a set A with respect to a universal set U is the set of all elements that belong to U but not to A. It is denoted by A'.
• Difference: The difference of two sets A and B is the set of all elements that belong to A but not to B. It is denoted by A \ B.
• Cartesian Product: The Cartesian product of two sets A and B is the set of all ordered pairs (a, b), where a belongs to A and b belongs to B. It is denoted by A × B.

#### Set Theory Definitions

• Set: A set is a collection of distinct objects.
• Element: An element is an object that belongs to a set.
• Subset: A set A is said to be a subset of a set B if every element of A also belongs to B. It is denoted by A ⊆ B.
• Power Set: The power set of a set A is the set of all possible subsets of A. It is denoted by P(A).
• Cardinality: The cardinality of a set A is the number of elements in A. It is denoted by |A|.
• Empty Set: The empty set is the set that has no elements. It is denoted by ∅.

### Set Theory Laws

• Commutative: A ∪ B = B ∪ A, A ∩ B = B ∩ A
• Associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C
• Distributive: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
• Identity: A ∪ ∅ = A, A ∩ U = A
• Complement: A ∪ A' = U, A ∩ A' = ∅
• De Morgan's: (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B'

### Set Theory Examples

#### Set Operations

Let A = {1, 2, 3} and B = {2, 4}.

• A ∪ B = {1, 2, 3, 4}
• A ∩ B = {2}
• A \ B = {1, 3}
• A' = {4}
• A × B = {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}

### Set Cardinality

If A = {1, 2, 3}, then |A| = 3.

## Set Types

• Finite Set: {1, 2, 3} is a finite set.
• Infinite Set: {1, 2, 3, ...} is an infinite set.
• Empty Set: The empty set, denoted by ∅ or {}, is the set that contains no elements.
• Singleton Set: {1} is a singleton set.
• Power Set: The power set of {1, 2} is {{}, {1}, {2}, {1, 2}}.
• Universal Set: The universal set, denoted by U, is the set that contains all possible elements.

# Number

Concept Definition Examples
Prime Numbers A positive integer greater than 1 that has no positive integer divisors other than 1 and itself. 2, 3, 5, 7, 11, 13
Composite Numbers A positive integer that is not prime. 4, 6, 8, 9
Prime Factorization The representation of a positive integer as a product of prime numbers. Prime factorization of 60 is 2 * 2 * 3 * 5
Greatest Common Divisor (GCD) The largest positive integer that divides both of the given integers without leaving a remainder. GCD of 24 and 36 is 12
Least Common Multiple (LCM) The smallest positive integer that is a multiple of both of the given integers. LCM of 6 and 8 is 24
Modular Arithmetic A system of arithmetic for integers, where numbers "wrap around" after reaching a certain value called the modulus. In modular arithmetic with modulus 5, 3 + 4 = 2, since 7 "wraps around" to 2 when divided by 5.

## Geometry Cheat Sheet

### Basic Concepts

Concept Definition
Point A location in space, represented by a dot.
Line A straight path that extends infinitely in both directions.
Ray A part of a line that has one endpoint and extends infinitely in one direction.
Line Segment A part of a line that has two endpoints.
Angle Two rays that share a common endpoint.
Plane A flat surface that extends infinitely in all directions.
Parallel Lines Two lines that never intersect and are always the same distance apart.
Perpendicular Lines Two lines that intersect at a right angle.

### Triangles

Triangle Type Description Properties
Equilateral All sides are equal Equal angles (60 degrees each)
Isosceles Two sides are equal Two equal angles
Scalene No sides are equal No equal angles
Right One angle is a right angle (90 degrees) Two sides are perpendicular

Square All sides are equal and all angles are right angles Diagonals are equal and bisect each other
Rectangle Opposite sides are equal and all angles are right angles Diagonals are equal and bisect each other
Parallelogram Opposite sides are parallel and equal in length Opposite angles are equal
Trapezoid One pair of opposite sides are parallel No sides are equal

## Circles

Circle Property Description
Radius The distance from the center of the circle to any point on the circle
Diameter The distance across the circle through its center
Circumference The distance around the circle
Area The amount of space inside the circle

## Trigonometry

Function Definition Periodicity
Sine (sin) The ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle.
Cosine (cos) The ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
Tangent (tan) The ratio of the length of the side opposite an angle to the length of the adjacent side in a right triangle. π
Cosecant (csc) The reciprocal of the sine function: 1/sin(x).
Secant (sec) The reciprocal of the cosine function: 1/cos(x).
Cotangent (cot) The reciprocal of the tangent function: 1/tan(x). π

## Trigonometric Identities

Identity Definition
Sine and Cosine sin²(x) + cos²(x) = 1
Pythagorean sin²(x) + cos²(x) = 1
Reciprocal sin(x) = 1/csc(x)
Quotient tan(x) = sin(x)/cos(x)
Even-Odd sin(-x) = -sin(x) and cos(-x) = cos(x)
Sum and Difference sin(x + y) = sin(x)cos(y) + cos(x)sin(y) and cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
Double-Angle sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x)

Matrix Operations Definition Example
Addition and Subtraction If A and B are matrices of the same size, then the sum/difference of A and B is the matrix obtained by adding/subtracting corresponding entries of A and B. A = [1 2; 3 4] and B = [5 6; 7 8], then A + B = [6 8; 10 12] and A - B = [-4 -4; -4 -4]
Scalar Multiplication If A is a matrix and k is a scalar, then the product of k and A is the matrix obtained by multiplying every entry of A by k. A = [1 2; 3 4] and k = 2, then kA = [2 4; 6 8]
Multiplication of Matrices If A is an m x n matrix and B is an n x p matrix, then the product of A and B is the m x p matrix obtained by multiplying the rows of A by the columns of B. If A = [1 2 3; 4 5 6] and B = [7 8; 9 10; 11 12], then AB = [58 64; 139 154]
Transpose If A is an m x n matrix, then the transpose of A is the n x m matrix obtained by interchanging rows and columns of A. If A = [1 2 3; 4 5 6], then A^T = [1 4; 2 5; 3 6]
Determinant The determinant of a square matrix A is a scalar value that can be computed from the entries of A. For a 2x2 matrix [a b; c d], the determinant is ad - bc. For larger matrices, the determinant can be computed using various methods. If A = [1 2; 3 4], then det(A) = -2
Inverse If A is a square matrix and det(A) is nonzero, then the inverse of A is the matrix A^-1 such that AA^-1 = I, where I is the identity matrix. If A = [1 2; 3 4], then A^-1 = [-2 1.5; 1 -0.5]

## Linear Algebra

### Basic Concepts

Concept Definition
Scalar An element of a field (e.g., a real number)
Vector An ordered set of scalars (e.g., a column or row of numbers)
Matrix An ordered set of vectors (e.g., a table of numbers)
Transpose A matrix where the rows become columns and the columns become rows

### Matrix Operations

Operation Definition
Addition The sum of two matrices with the same dimensions is a matrix where each element is the sum of the corresponding elements of the two matrices
Scalar Multiplication A matrix multiplied by a scalar is a matrix where each element is the product of the scalar and the corresponding element of the matrix
Multiplication The product of two matrices A and B is a matrix C where each element cij is the dot product of the ith row of A and the jth column of B
Inverse A matrix A has an inverse A^-1 if the product of A and A^-1 is the identity matrix

### Eigenvalues and Eigenvectors

Concept Definition
Eigenvalue A scalar λ that satisfies the equation A*v = λ*v for some nonzero vector v, where A is a square matrix
Eigenvector A nonzero vector v that satisfies the equation A*v = λ*v for some scalar λ, where A is a square matrix
Eigendecomposition A diagonalization of a square matrix A as A = V*D*V^-1, where V is a matrix of eigenvectors of A and D is a diagonal matrix of the corresponding eigenvalues

## Calculus

### Derivatives

Function Derivative
Constant function: c f'(x) = 0
Power function: xn f'(x) = nxn-1
Exponential function: ex f'(x) = ex
Logarithmic function: loga(x) f'(x) = 1 / (x * ln(a))
Trigonometric functions:
Sine function: sin(x) f'(x) = cos(x)
Cosine function: cos(x) f'(x) = -sin(x)
Tangent function: tan(x) f'(x) = sec2(x)

### Integrals

Function Integral
Power function: xn ∫ xn dx = (xn+1)/(n+1) + C
Exponential function: ex ∫ ex dx = ex + C
Logarithmic function: ln(x) ∫ ln(x) dx = x ln(x) - x + C
Trigonometric functions:
Sine function: sin(x) ∫ sin(x) dx = -cos(x) + C
Cosine function: cos(x) ∫ cos(x) dx = sin(x) + C
Tangent function: tan(x) ∫ tan(x) dx = ln|sec(x)| + C

### Rules

Rule Explanation
Product rule (f * g)' = f' *g' g' * f
Quotient rule (f / g)' = (f' * g - g' * f) / g2
Chain rule f'(g(x)) = f'(u) * u'(x), where u = g(x)
Integration by substitution ∫ f(g(x)) * g'(x) dx = ∫ f(u) du, where u = g(x)
Integration by parts ∫ u dv = u * v - ∫ v du

### Common derivatives and integrals

Function Derivative Integral
sin(x) cos(x) -cos(x) + C
cos(x) -sin(x) sin(x) + C
ex ex ex + C
ln(x) 1 / x x ln(x) - x + C
1 / (1 + x2) -2x / (1 + x2)2 arctan(x) + C

### Important concepts

Concept Explanation
Limits The value a function approaches as the input approaches a certain value.
Continuity A function is continuous if it has no breaks, holes, or jumps.
Differentiability A function is differentiable if it has a derivative at every point in its domain.
Mean value theorem If a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f'(c) = (f(b) - f(a)) / (b - a).
Fundamental theorem of calculus If f(x) is continuous on [a,b], then ∫ab f(x) dx = F(b) - F(a), where F(x) ab f(x) dx
Indefinite integral The antiderivative of a function, which gives a family of functions that differ by a constant.
Definite integral The area under a curve between two points on the x-axis.
Area between curves The area between two curves can be found by subtracting the integral of the lower curve from the integral of the upper curve.
Volume by slicing The volume of a solid can be found by slicing it into thin pieces and adding up the volumes of those pieces.
Volume by shells The volume of a solid can be found by slicing it into thin shells and adding up the volumes of those shells.

### Tips and tricks

• Always simplify before taking a derivative or integral.
• Make sure to use parentheses when taking derivatives and integrals of composite functions.
• Remember the power rule for integrals: ∫ xn dx = (xn+1)/(n+1) + C.
• Don't forget the constant of integration when evaluating indefinite integrals.
• When using integration by parts, choose u and dv so that the integral of v du is easier to evaluate than the original integral.

With this cheat sheet, you should have a solid foundation for working with calculus. Good luck with your studies!

## Fourier Analysis Cheat Sheet

### Introduction

Fourier analysis is a mathematical technique that allows us to represent a function as a sum of sine and cosine waves of different frequencies. It has many applications in signal processing, image analysis, and physics.

### Fourier Series

A Fourier series is a representation of a periodic function as a sum of sine and cosine functions with different frequencies. The general form of a Fourier series is:

f(x) = a0 + ∑n=1 (an cos(nx) + bn sin(nx))

The coefficients an and bn can be computed using the following formulas:

Coefficient Formula
a0 a0 = (1/2π) ∫π f(x) dx
an an = (1/π) ∫π f(x) cos(nx) dx
bn bn = (1/π) ∫π f(x) sin(nx) dx

Here, π is the value of pi (3.14159...).

### Fourier Transform

The Fourier transform is a generalization of the Fourier series to non-periodic functions. It allows us to represent a function as a sum of sine and cosine waves of all frequencies, not just integer multiples of a base frequency.

The Fourier transform of a function f(x) is given by:

F(k) = ∫-∞ f(x) e-2πikx dx

The inverse Fourier transform allows us to recover the original function from its Fourier transform:

f(x) = (1/√2π) ∫-∞ F(k) e2πikx dk

### Properties of Fourier Transforms

The Fourier transform has many important properties that make it a powerful tool for analyzing functions. Some of the most important properties are:

Property Formula
Linearity F(a f(x) + b g(x)) = a F(f(x)) + b F(g(x))
Shift F(f(x - a)) = e-2πika F(f(x))
Scaling F(f(ax)) = (1/|a|) F(f(x/a))
Derivative F(df/dx) = 2πi k F(f(x))
Convolution F(f(x) * g(x)) = F(f(x)) F(g(x))
Parseval's Theorem -∞ |f(x)|2 dx = ∫-∞ |F(k)|2 dk

Here, a and b are constants, and f(x) and g(x) are functions.

## Applications

Fourier analysis has many applications in signal processing, image analysis, and physics. Some examples of its applications are:

• Audio signal processing, such as filtering and compression
• Image analysis, such as image compression and edge detection
• Quantum mechanics, where the wave function of a particle is represented as a superposition of sine and cosine waves

## Conclusion

Fourier analysis is a powerful mathematical technique that allows us to represent a function as a sum of sine and cosine waves of different frequencies. It has many applications in signal processing, image analysis, and physics, and is an essential tool for any engineer or scientist.

## Laplace Transforms

Laplace transforms are a mathematical technique used to solve differential equations by converting them into algebraic equations. They are widely used in engineering, physics, and other fields. This cheat sheet provides some common Laplace transform pairs and properties.

### Laplace Transform Pairs

The Laplace transform of a function f(t) is denoted by F(s) and is defined as:

F(s) = ∫0 f(t) e-st dt

Here are some common Laplace transform pairs:

f(t) F(s)
1 1/s
tn n!/sn+1
eat 1/(s-a)
sin(bt) b/(s2+b2)
cos(bt) s/(s2+b2)
f(t-a)u(t-a) e-asF(s)
u(t-a) 1/s e-as

Here, n is a positive integer, a and b are constants, and u(t-a) is the unit step function defined as:

u(t-a) = {0, t<a; 1, t>=a}

### Laplace Transform Properties

The Laplace transform has several properties that can be used to simplify calculations. Here are some common properties:

Property Laplace Transform
Linearity a*f(t) + b*g(t) → a*F(s) + b*G(s)
Time Shifting f(t-a)u(t-a) → e-asF(s)
Frequency Shifting eatf(t) → F(s-a)
First Derivative f'(t) → sF(s) - f(0)
Higher Derivatives f(n)(t) → snF(s) - sn-1f(0)
Integration 0t f(τ) dτ → F(s)/(s)
Initial Value Theorem lims→∞ sF(s) = f(0)
Final Value Theorem lims→0 sF(s) = limt→∞ f(t)
Convolution f(t)*g(t) → F(s)G(s)
Derivative in s tnF(s) → (-1)ndnF(s)/dsn
Initial Value F(s) = ∫0 f(t) e-st dt = L(f(t))

These properties can be used to simplify Laplace transforms and solve differential equations in a more efficient way.

## Conclusion

Laplace transforms are a powerful tool for solving differential equations in a wide range of applications. This cheat sheet provides some common Laplace transform pairs and properties that can be used to simplify calculations. It is important to have a good understanding of these concepts in order to apply them effectively in practice.

## Z-Transforms

The Z-transform is a mathematical tool used to analyze discrete-time signals and systems. It is closely related to the Laplace transform, which is used to analyze continuous-time signals and systems. This cheat sheet provides some common Z-transform pairs and properties that can be used to simplify calculations.

### Z-Transform Pairs

The Z-transform of a discrete-time signal x[n] is defined as:

where z is a complex variable. The inverse Z-transform is defined as:

where C is a closed contour in the complex plane that encloses all the poles of X(z). The Z-transform pairs are shown in the following table:

Signal x[n] Z-Transform X(z)
δ[n] 1
an&u;[n] Σn=0 anz-n, |z|>|a|
n&u;[n] Σn=0 z-n, |z|>1
sin(w0n)&u;[n] Σn=-\infin; (-1)njnz-n, |z|=1
cos(w0n)&u;[n] Σn=-\infin; z-ncos(w0n)
ea0n&u;[n] Σn=0 ea0nz-n, |z|>|a0|

where Σ denotes the sum over all integer values of n, ∞ denotes infinity, and u[n] is the unit step function.

### Z-Transform Properties

The Z-transform has several useful properties that can be used to simplify calculations. The most common properties are:

#### Linearity

The Z-transform is a linear operator, which means that it satisfies the following property:

where a and b are constants.

#### Shift

The Z-transform of a shifted signal can be expressed as:

where k is a constant.

#### Convolution

The Z-transform of a convolution can be expressed as:

where * denotes convolution.

#### Differentiation

The Z-transform of a differentiated signal can be expressed as:

#### Initial Value Theorem

The initial value theorem states that:

where x is the initial value of the signal.

#### Final Value Theorem

The final value theorem states that:

where x[n] is a bounded signal.

### Conclusion

The Z-transform is a powerful tool for analyzing discrete-time signals and systems. By using the Z-transform pairs and properties provided in this cheat sheet, you can simplify calculations and gain a deeper understanding of the behavior of discrete-time signals and systems.

## Convolution

Convolution is a mathematical operation used in signal processing and many other fields. It is used to combine two functions to produce a third function that describes how one of the original functions modifies the other.

Convolution can be represented in various forms, including continuous-time convolution, discrete-time convolution, and matrix convolution. In this cheat sheet, we will focus on discrete-time convolution.

### Discrete-time Convolution

The discrete-time convolution of two sequences x[n] and h[n] is defined as:

where y[n] is the resulting sequence.

The discrete-time convolution is also denoted as x[n]*h[n].

### Properties of Convolution

Convolution has several important properties that make it a powerful tool for signal processing.

Property Formula Explanation
Commutativity x[n]*h[n] = h[n]*x[n] The order of the sequences does not affect the result of convolution.
Associativity (x[n]*h1[n])*h2[n] = x[n]*(h1[n]*h2[n]) Convolution is associative, which means that the grouping of the sequences does not affect the result of convolution.
Distributivity x[n]*(h1[n]+h2[n]) = x[n]*h1[n] + x[n]*h2[n] Convolution is distributive over addition.
Linearity a*x1[n] + b*x2[n] * h[n] = a*(x1[n]*h[n]) + b*(x2[n]*h[n]) Convolution is linear, which means that it satisfies the superposition principle.
Shift Invariance If y[n] = x[n]*h[n], then y[n-k] = x[n-k]*h[n-k] If the input sequence is shifted, the output sequence is also shifted by the same amount.
Convolution with Delta Function x[n]*δ[n-k] = x[n-k] Convolution with a delta function produces a shifted version of the input sequence.
Convolution with Unit Step Function x[n]*u[n-k] = ∑i=k n x[i] Convolution with a unit step function produces a running sum of the input sequence.

### Conclusion

Convolution is a powerful mathematical tool thatis widely used in signal processing, image processing, and many other fields. Understanding the properties of convolution can help you analyze and manipulate signals with ease. With this cheat sheet, you can quickly reference the formula and properties of discrete-time convolution.

## Sampling Theorem Cheat Sheet

The sampling theorem, also known as the Nyquist-Shannon sampling theorem, is a fundamental concept in signal processing that states that a continuous-time signal can be reconstructed from its samples, provided that the sampling rate is greater than or equal to twice the maximum frequency of the signal.

In this cheat sheet, we will summarize the key points of the sampling theorem.

### The Sampling Theorem

The sampling theorem states that:

• A continuous-time signal with a maximum frequency of fm can be reconstructed from its samples, if the sampling rate is at least 2fm.
• The sampling rate required for perfect reconstruction is called the Nyquist rate.
• The Nyquist rate is given by:

where fs is the sampling rate.

### Aliasing

When the sampling rate is lower than the Nyquist rate, aliasing occurs. Aliasing is a phenomenon where a high-frequency signal appears as a low-frequency signal in the reconstructed signal. Aliasing can lead to errors in signal processing and must be avoided.

### Reconstruction

There are various methods for reconstructing a continuous-time signal from its samples, including:

• Zero-order hold (ZOH) interpolation
• First-order hold (FOH) interpolation
• Sinc interpolation

### Conclusion

The sampling theorem is a fundamental concept in signal processing that provides the theoretical basis for digital signal processing. With this cheat sheet, you can quickly reference the key points of the sampling theorem and ensure that you are sampling your signals correctly to avoid aliasing and reconstruct them accurately.

## Discrete Mathematics Cheat Sheet

Discrete mathematics is a branch of mathematics that deals with discrete objects and structures, as opposed to continuous objects like real numbers and continuous functions. It is used in computer science, engineering, and many other fields. This cheat sheet summarizes some of the key concepts in discrete mathematics.

### Propositional Logic

Propositional logic deals with propositions or statements that are either true or false. The following table summarizes some of the key logical operators:

Operator Symbol Example
Negation ~ ~p (not p)
Conjunction & p & q (p and q)
Disjunction | p | q (p or q)
Implication -> p -> q (if p then q)
Equivalence <-> p <-> q (p if and only if q)

### Set Theory

Set theory deals with sets, which are collections of objects. The following table summarizes some of the key set operations:

Operation Symbol Example
Union A ∪ B (elements in A or B or both)
Intersection A ∩ B (elements in both A and B)
Complement ' A' (elements not in A)
Subset A ⊆ B (A is a subset of B)
Universal Set U U (set of all possible elements)

### Graph Theory

Graph theory deals with graphs, which are collections of vertices (nodes) and edges (lines connecting vertices). The following table summarizes some of the key graph concepts:

Concept Definition
Degree Number of edges incident to a vertex
Path Sequence of edges connecting vertices
Cycle Path that starts and ends at the same vertex
Connected Graph where there is a path between any two vertices
Tree Connected acyclic graph
Spanning Tree Tree that includes all vertices in a graph

### Number Theory

Number theory deals with the properties of integers. The following table summarizes some of the key number theory concepts:

Concept Definition
Divisibility One integer is divisible by another if it is a multiple of that integer
Prime Number Integer greater than 1 that is only divisible by 1 and itself
Greatest Common Divisor Largest integer that divides two given integers
Modular Arithmetic Arithmetic with remainders, where two integers are considered equivalent if they have the same remainder when divided by a given integer (the modulus)
Modular Inverse Integer that, when multiplied by a given integer modulo m, yields 1

These are some of the key concepts in discrete mathematics, and there are many more. However, this cheat sheet should provide a useful summary for anyone studying the subject.

## Graph Theory

Graph theory is the study of graphs, which are mathematical structures that model pairwise relationships between objects. The following table summarizes some of the key concepts in graph theory:

Concept Definition
Graph Collection of vertices (or nodes) and edges that connect them
Directed Graph Graph where edges have a direction (represented by arrows)
Undirected Graph Graph where edges have no direction
Weighted Graph Graph where edges have weights (numbers that represent the "cost" or "distance" of traversing that edge)
Complete Graph Undirected graph where every pair of vertices is connected by an edge
Path Sequence of vertices connected by edges
Cycle Path that starts and ends at the same vertex
Connected Graph where there is a path between any two vertices
Tree Connected acyclic graph
Spanning Tree Tree that includes all vertices in a graph

In addition to these concepts, there are many algorithms and properties that are important in graph theory. Some of the most common ones are:

Algorithm/Property Description
Breadth-First Search Algorithm for traversing a graph by exploring all vertices at a given distance from the starting vertex before moving on to vertices that are further away
Depth-First Search Algorithm for traversing a graph by exploring as far as possible along each branch before backtracking
Shortest Path Algorithm for finding the shortest path between two vertices in a weighted graph
Minimum Spanning Tree Algorithm for finding the minimum-weight spanning tree of a weighted graph
Eulerian Graph Graph where there is a path that visits every edge exactly once
Hamiltonian Graph Graph where there is a path that visits every vertex exactly once

These are some of the key concepts and algorithms in graph theory, and there are many more. However, this cheat sheet should provide a useful summary for anyone studying the subject.

## Game Theory

Game theory is the study of decision-making in situations where two or more individuals or groups have conflicting interests. The following table summarizes some of the key concepts in game theory:

Concept Definition
Game A set of players, a set of strategies available to each player, and a set of payoffs that each player receives for each possible combination of strategies
Players The individuals or groups involved in a game
Strategies The options available to each player
Payoffs The rewards or penalties received by each player for each combination of strategies played
Nash Equilibrium A set of strategies where no player can improve their payoff by unilaterally changing their strategy, assuming the other players' strategies remain unchanged
Dominant Strategy A strategy that is always the best choice for a player, regardless of what the other players do
Prisoner's Dilemma A famous game in which two individuals are arrested for a crime and must decide whether to confess or remain silent, with the payoffs dependent on the other player's decision
Tragedy of the Commons A situation in which individuals or groups exploit a shared resource for their own benefit, leading to depletion or degradation of the resource
Zero-Sum Game A game in which the total payoff for all players is constant, so that any gain for one player is matched by an equal loss for another player

There are many other concepts and strategies used in game theory, including mixed strategies, repeated games, and bargaining theory. These can be used to analyze a wide range of situations, from economic competition to international diplomacy to social interactions. By understanding the principles of game theory, individuals can make more informed decisions in a variety of contexts.

## Optimization

Optimization is the process of finding the best solution to a problem, given a set of constraints. The following table summarizes some of the key concepts and techniques used in optimization:

Concept Definition
Objective Function The function that is being maximized or minimized
Constraints The conditions that must be satisfied for the solution to be valid
Feasible Region The set of all points that satisfy the constraints
Optimal Solution The best solution that satisfies the constraints and maximizes or minimizes the objective function
Linear Programming A method for optimizing a linear objective function subject to linear constraints
Nonlinear Programming A method for optimizing a nonlinear objective function subject to nonlinear constraints
Gradient Descent A method for finding the minimum of a function by iteratively adjusting the parameters in the direction of steepest descent
Newton's Method A method for finding the minimum of a function by using the second derivative to estimate the curvature and adjust the parameters accordingly
Quadratic Programming A method for optimizing a quadratic objective function subject to linear constraints
Convex Optimization A class of optimization problems where the objective function and constraints satisfy certain properties that guarantee a unique optimal solution

There are many other concepts and techniques used in optimization, including integer programming, dynamic programming, and stochastic programming. These can be used to solve a wide range of problems, from maximizing profits to minimizing costs to optimizing resource allocation. By understanding the principles of optimization, individuals can make more informed decisions in a variety of contexts.