Chapter 2 notes

Conditional Statement

Form: $p\rightarrow q$ = 'If p, then q'

Truth Rule for Conditionals

A conditional statement is only false when p is true and q is false.

Vacuously True

A statement is considered true by default if p is false.

Logical Equivalence of Conditionals

If p then q is equivalent to Not p or q

Negation of a Conditional

Negation of $p\rightarrow q$ is $p\land\neg q$, meaning 'p is true AND q is false'.

Contrapositive

The contrapositive of $p\rightarrow q$ is $\neg q\rightarrow\neg p$ and is logically equivalent to the original conditional.

Converse

The converse of $p\rightarrow q$ is $q\rightarrow p$.

Inverse

The inverse of $p\rightarrow q$ is $\neg p\rightarrow\neg q$.

'Only If' in Conditionals

'p only if q' means $p\rightarrow q$ and has an equivalent contrapositive $\neg q\rightarrow\neg p$.

Biconditional Statement

A biconditional statement is true if both statements have the same truth value, indicating that either both are true or both are false. It can be expressed in the form "p if and only if q" and is often denoted as "p ↔ q."

Sufficient Condition

A condition A guarantees B, described as 'If A then B'.

Necessary Condition

A condition B must be true for A to be true, described as 'If not B then not A'.

Order of Operations in Logic

Order of operations is: NOT first, then AND, OR, and finally if-then.