Sets
What is a set?
A set is made up of elements.
Cardinality
The cardinality (size) of a set is the number of elements in it.|A| = 4 (there are 4 elements in A, so the cardinality is 4)
Intersections
The intersection is defined as elements that are in both sets. Symbol ∩: “intersects” (and)A ∩ B = {x : x ∈ A and x ∈ B}If there are no elements in common, the answer is the empty set ∅. The cardinality of the empty set |∅| = 0.
Unions
The union is defined as elements that are in either set. Symbol ∪: “union” (or)A ∪ B = {x ∈ A or x ∈ B}.
Visualizing sets
Venn diagrams
Inclusion-exclusion formula
Inclusion-exclusion formula: |A ∪ B| = |A| + |B| − |A ∩ B|
Numbers
Integers and rational numbers
Some real numbers terminate, and some do not. The number π = 3.14159… is irrational, it does not repeat after the decimal point.
Absolute value
The absolute value of a number x, |x|, is the distance from x to 0.
Intervals and Interval Notation
Closed intervals [2, 3.1]Open intervals (5, 8)Half-open intervals (2, 3], [20, 20.3)
Sigma Notation
Sigma notation (Σ)
distributive property: a(b + c) = ab + accommutative property: a + b = b + a
Cartesian Plane
Axes and quadrants
x−axisy−axisfirst quadrantsecond quadrantthird quadrantfourth quadrant
Pythagorean theorem
Derivation using point-slope form
Slope-intercept form
If L has slope m, and hits the y-axis at (0, b), then y = mx + b is an equation for L, where m is the slope and b is the y-intercept.
Point-Slope Formula for Lines
y − y0 = m(x − x0) Point-slope formy = mx + b Slope-intercept form
Functions
f : A → B
Tangent Lines
The Slope of a Graph at a Point
The slope of the tangent line gives the instantaneous rate of change. This is also called the derivative of the function at that point, or f(a).
The Derivative Function
Derivative formula: f(a) = lim (f(a + h) − f(a)) /h
Fast Growth, Slow Growth
Integer Exponents
Multiplication rule: x^n * x^m = x^(n+m)Power to a power: (xn)m = x^nmProduct to a power: (x * y)^n = x^n * y^nFraction to a power: (x/y)n=xn / y^nDivision and negative powers: x^n / x^m= x^(n−m)
How Logarithms and Exponents Are Related
b^x = N “exponential form”x = log b(N) “logarithmic form”
Product rule: log(xy) = log(x) + log(y)Quotient rule: log(x/y) = log(x) − log(y)Power and root rule: log(x^n) = n log(x)
Basic Probability Definitions
probability—the degree of belief in the truth or falsity of a statementRange of uncertainty from 0 to 1P(x) probability of x∼x negation of statement xjoint probability—probability that two separate events with separate probability distributions are both true.P(A and B) is written P(A, B), and read “the joint probability of A and B” or “the probability that A is true and B is true.”
Problem Solving Methods
Permutations and Combinations
permutation—order matters, n! / (n-m)!combination—order does not matter, n! / (m! * (n-m)!)
Using Factorial and “M Choose N”
(m n) = m! / ((m − n)! · n!)
The Sum Rule, Conditional Probability, and the Product Rule
P(A) = P(A, B1) + P(A, B2) + … + P(A, Bn)P(A | B) = (relevant outcomes) / (total outcomes remaining in universe, when B is true)P(A | B) = P(A, B) / P(B)
Bayes’ Theorem
P(A | B) = P(B | A) * P(A) / P(B)Technical vocabulary of Bayesian inverse probability: posterior probability = likelihood * prior probability / marginal probability
posterior probability—probability after new data is observedprior probability—probability before any data is observed or before new data is observedlikelihood—standard forward probability of data given parametersmarginal probability—probability of the data
The Binomial Theorem and Bayes’ Theorem
Binomial theorem used when there are two possible outcomes—a success or a non-success, for example, flipping a coin—heads are a success, binary outcome.Not limited to fair coins, where the probability of success is 0.5. Probability can be any value > 0 and < 1.Probability of s successes in n trials, when probability of 1 success is p: (n s) * p^s * (1 − p)^(n−s)
where n is the number of independent trials (with replacement), s is the number of successes,and p is the probability of one success
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