Logical connectives are operators used to combine or modify propositions to form compound propositions.Below is a complete and exam-oriented explanation of all standard logical connectives, along with truth tables.
Negation reverses the truth value of a proposition.
| p | ¬p |
|---|---|
| T | F |
| F | T |
A conjunction is true only when both propositions are true.
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
A disjunction is true if at least one proposition is true.(This is inclusive OR.)
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
“If p, then q”It is false only when p is true and q is false.
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
⚠️ Important exam note:When p is false, the implication is always true (called vacuous truth).
“p if and only if q”It is true when both propositions have the same truth value.
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |