Statements and Connectives + Truth Tables
Logical statements form the backbone of mathematical reasoning through five key connectives. A negation (~) simply reverses truth values - what was true becomes false and vice versa. The conjunction (∧) represents "AND" logic, requiring both statements to be true for the result to be true.
The disjunction (∨) represents "OR" logic, where only one statement needs to be true for the result to be true. With implication (→), a statement "if p then q" is only false when the hypothesis is true but the conclusion is false. The biconditional (↔) represents "if and only if" logic, true only when both statements have matching truth values.
When working with a conditional statement (p → q), three related statements emerge: the converse (q → p), the inverse (~p → ~q), and the contrapositive (~q → ~p). Importantly, a conditional statement is logically equivalent to its contrapositive, while the converse is equivalent to the inverse.
Remember This! When creating truth tables for complex statements like (p ∧ q) ∨ ~q, always work from the inside out, evaluating the simplest components first before combining them into the final result.