Math 30-2 Diploma flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Statement

A declarative sentence that is either true or false, but not both. Questions, commands, and expressions of opinion do not count as statements in logic.

Negation

The logical opposite of a statement, written ~p. If the original statement is true, its negation is false, and if the original is false, the negation is true.

Conjunction

A compound statement formed by joining two statements with AND, written p ∧ q. It is true only when both component statements are true and false otherwise.

Disjunction

A compound statement formed by joining two statements with OR, written p ∨ q. It is false only when both component statements are false and true otherwise.

Conditional statement

A statement of the form if p then q, written p → q. It is false only when p is true and q is false; every other combination makes it true.

Contrapositive

The statement ~q → ~p formed from p → q by negating and swapping the parts. It is always logically equivalent to the original conditional statement.

Converse

The statement q → p formed by swapping the hypothesis and conclusion of p → q. A converse is not automatically equivalent to its original conditional statement.

Counterexample

A single specific case in which the premises are true but the conclusion is false. One counterexample is enough to disprove a general claim or argument.

Theoretical probability

The ratio of favourable outcomes to total equally likely outcomes, P(A) = favourable / total. It describes what should happen in theory rather than what was observed.

Complement rule

The probability that event A does not occur equals 1 minus the probability that A occurs, written P(A') = 1 − P(A).

Mutually exclusive events

Two events that cannot both happen in the same trial, so P(A and B) = 0. For such events the addition rule simplifies to P(A or B) = P(A) + P(B).

Independent events

Two events where the occurrence of one does not change the probability of the other. They satisfy P(A and B) = P(A) × P(B).

Conditional probability

The probability that event B occurs given that event A has already occurred, computed as P(B|A) = P(A and B) / P(A), where P(A) is not zero.

Addition rule

For any two events, P(A or B) = P(A) + P(B) − P(A and B). The overlap is subtracted so outcomes counted in both events are not double-counted.

Expected value

The long-run average outcome found by multiplying each possible value by its probability and summing the products. It need not equal any single actual outcome.

Odds in favour

A ratio comparing favourable outcomes to unfavourable outcomes, written a:b. The corresponding probability is a divided by the sum a + b.

Function

A relation in which each input value in the domain is paired with exactly one output value. Different inputs may share the same output without breaking the rule.

Domain and range

The domain is the complete set of input (x) values a relation may use; the range is the complete set of resulting output (y) values.

Vertical line test

A graph represents a function only if no vertical line intersects the graph at more than one point. Two points on one vertical line mean one input has two outputs.

Function notation

The symbol f(x) names the output of function f for input x. For example, f(3) = 7 means the function produces 7 when the input is 3.

Slope

The rate of change of a linear relation, m = (y₂ − y₁)/(x₂ − x₁). It measures vertical change per unit of horizontal change between two points.

Vertex of a parabola

The highest or lowest point of a quadratic graph. In y = a(x − h)² + k the vertex is (h, k) and the axis of symmetry is x = h.

Exponential growth and decay

A model y = a·bˣ where a is the starting value. If b > 1 the quantity grows; if 0 < b < 1 the quantity decays toward zero.

Inverse of a relation

The relation formed by swapping the x and y coordinates of every ordered pair. Its graph is a reflection of the original graph across the line y = x.

Radian measure

An angle measure based on arc length, where a full circle is 2π radians. Degrees convert to radians by multiplying by π/180.

Arc length formula

For a circle of radius r and central angle θ in radians, the arc length is s = rθ. The angle must be in radians for the formula to work.

Sector area formula

The area of a circular sector with radius r and central angle θ in radians is A = ½r²θ, a fraction of the full circle area πr².

Pythagorean identity

For any angle θ, sin²θ + cos²θ = 1. This follows from the unit circle, where the point (cos θ, sin θ) lies on a circle of radius one.

Unit circle coordinates

On a circle of radius one centred at the origin, the point at angle θ has coordinates (cos θ, sin θ). Cosine gives the x-value and sine the y-value.

Amplitude and period

For y = a·sin(b(x − c)) + d, the amplitude is |a| and the period is 2π/|b|. Amplitude is half the distance from maximum to minimum.

Sine law

The ratio a/sin A = b/sin B = c/sin C holds in any triangle. It is used with two angles and a side, or two sides and a non-included angle.

Cosine law

The relation c² = a² + b² − 2ab·cos C holds in any triangle. It is used with two sides and the included angle, or with all three sides.