Truth Tables in Propositional Logic: Complete Guide
A truth table is a mathematical table used in propositional logic, computer science, and boolean algebra to compute the functional values of logical expressions for each combination of truth values of their variables.
The Five Fundamental Logical Operators
- 1Negation ($\neg P$): Flips true to false and false to true.
- 2Conjunction ($P \land Q$, AND): True only if both $P$ and $Q$ are true.
- 3Disjunction ($P \lor Q$, OR): True if at least one operand is true.
- 4Conditional ($P \implies Q$, If-Then): False only when a true antecedent leads to a false consequent ($T \implies F$). Vacuously true whenever $P$ is false.
- 5Biconditional ($P \iff Q$, If and only if): True when both operands share the identical truth value ($T \leftrightarrow T$ or $F \leftrightarrow F$).
Comprehensive Master Truth Table
| $P$ | $Q$ | $\neg P$ | $P \land Q$ (AND) | $P \lor Q$ (OR) | $P \oplus Q$ (XOR) | $P \implies Q$ (IMPLIES) | $P \iff Q$ (IFF) |
|---|---|---|---|---|---|---|---|
| T | T | F | T | T | F | T | T |
| T | F | F | F | T | T | F | F |
| F | T | T | F | T | T | T | F |
| F | F | T | F | F | F | T | T |