Complete Logic & Proof Study Guide

Statement

A sentence that is either true or false (not both).

Truth Value

Whether a statement is true or false.

Negation

The opposite meaning and truth value of a statement.

Compound Statement

Two or more statements joined by 'and' or 'or'.

Conditional Statement

A statement of the form 'If P, then Q' where P is the hypothesis and Q is the conclusion.

Related Conditionals

Converse, Inverse, and Contrapositive of a conditional statement.

Converse

The statement formed by swapping the hypothesis and conclusion of a conditional statement.

Inverse

The statement formed by negating both the hypothesis and conclusion of a conditional statement.

Contrapositive

The statement formed by negating both the hypothesis and conclusion of the converse.

Biconditional Statement

A statement that is true when both parts have the same truth value, usually written as 'P if and only if Q'.

Inductive Reasoning

A method of reasoning that involves observing patterns and forming conjectures based on specific examples.

Deductive Reasoning

A method of reasoning from the general to the specific, using established facts or premises.

Law of Detachment

If 'If P, then Q' is true and P is true, then Q must be true.

Law of Syllogism

If 'If P, then Q' and 'If Q, then R' are true, then 'If P, then R' is true.

Algebraic Properties

Laws governing mathematical operations, including addition, subtraction, multiplication, and division.

Fundamental Properties

Basic properties of equality pertaining to reflexive, symmetric, transitive, and substitution.

Operational Properties

Properties related to algebraic operations such as distributive, commutative, and associative.

Postulate

A statement accepted as true without proof.

Midpoint

If M is the midpoint of AB, then AM = MB (Equality) and AM ≅ MB (Congruence).

Counterexample

An example that disproves a conjecture.