Math 30-2 Diploma cheat sheet

A statement is a sentence that is either true or false; questions, commands, and opinions are not statements.

Negation (~p) reverses truth value: if p is true, ~p is false, and vice versa.

Conjunction (p AND q) is true only when both p and q are true; disjunction (p OR q) is false only when both are false.

Conditional p → q is false only in the case where p is true and q is false; it is true in all other cases.

The converse of p → q is q → p; the inverse is ~p → ~q; the contrapositive is ~q → ~p. A conditional and its contrapositive are logically equivalent.

Biconditional p ↔ q is true exactly when p and q have the same truth value.

De Morgan's laws: ~(p AND q) ≡ (~p OR ~q) and ~(p OR q) ≡ (~p AND ~q).

A valid deductive argument has a conclusion that must follow from the premises; an inductive argument generalizes from specific cases and is only probable.

A counterexample is a single case where the premises are true but the conclusion is false, which disproves a general claim.

Probability of an event: P(A) = (number of favourable outcomes) / (total number of equally likely outcomes).

0 ≤ P(A) ≤ 1; P(impossible event) = 0 and P(certain event) = 1.

Complement rule: P(A') = 1 − P(A).

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events P(A and B) = 0.

Multiplication rule for independent events: P(A and B) = P(A) × P(B).

Conditional probability: P(B|A) = P(A and B) / P(A), provided P(A) ≠ 0.

Two events are independent when P(A and B) = P(A) × P(B), equivalently P(B|A) = P(B).

Odds in favour of A are (favourable) : (unfavourable); probability from odds a:b is P = a/(a+b).

Expected value = Σ (value × probability of that value); it need not equal any actual outcome.

A relation is a set of ordered pairs; a function assigns exactly one output value to each input value in its domain.

The vertical line test: a graph represents a function only if no vertical line crosses it more than once.

Domain is the set of all input (x) values; range is the set of all output (y) values.

Function notation: y = f(x); f(3) means the output when the input is 3.

A linear function y = mx + b has slope m (rate of change) and y-intercept b; slope m = (y₂ − y₁)/(x₂ − x₁).

Quadratic y = a(x − h)² + k has vertex (h, k); the axis of symmetry is x = h; a > 0 opens upward, a < 0 opens downward.

Exponential y = a·bˣ has growth when b > 1 and decay when 0 < b < 1; the parameter a is the initial value at x = 0.

A relation's inverse swaps x and y in each ordered pair; the inverse of a function is also a function only if the original passes the horizontal line test.

Transformations: y = a·f(b(x − h)) + k stretches by a, compresses by b, shifts right h and up k.

Angles can be measured in degrees or radians; 180° = π radians, so degrees × (π/180) = radians.

Arc length s = rθ and sector area A = ½r²θ, where θ is measured in radians.

In a right triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Pythagorean identity: sin²θ + cos²θ = 1; also tan θ = sin θ / cos θ.

On the unit circle, a point at angle θ has coordinates (cos θ, sin θ); sine is the y-coordinate and cosine the x-coordinate.

The sine and cosine functions have period 2π (360°) and amplitude 1; tangent has period π (180°).

For y = a·sin(b(x − c)) + d: amplitude = |a|, period = 2π/|b|, vertical shift = d, phase shift = c.

Sine law: a/sin A = b/sin B = c/sin C; use it when given two angles and a side, or two sides and a non-included angle.

Cosine law: c² = a² + b² − 2ab·cos C; use it when given two sides and the included angle, or all three sides.