Logical Connectives
definitions
discrete-math
logic
propositional-logic
Statements
A statement (or proposition) is a sentence that is either true or false, but not both. Represented by variables \(p, q, r, \dots\)
Basic connectives
| Connective | Symbol | Alt. symbol |
|---|---|---|
| not | \(\sim\) | \(\neg\) |
| and | \(\land\) | |
| or (inclusive) | \(\lor\) |
\(p \lor q\) is true when \(p\) is true, \(q\) is true, or both (“inclusive or”).
Exclusive or (\(\oplus\))
“One or the other, but not both”:
\[p \oplus q \equiv (p \lor q) \land \sim(p \land q)\]
| \(p\) | \(q\) | \(p \oplus q\) |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
Order of operations
- \(\sim\) first.
- \(\land, \lor\) equal second — parenthesise, else left to right.
Truth tables
A truth table for a statement form with \(n\) variables has \(2^n\) rows. Build in standard form (systematically covering every combination, e.g. TTT, TTF, TFT, …) so it’s easy to verify completeness.
See also
- logical-equivalence-laws
- conditional-statements
- practice-problems — §1 Logical Form, with full worked solutions