Truth Tables
A TRUTH TABLE shows how the TRUTH or FALSITY of a compound statement depends on the truth values of its components. For each combination of input truth values, you compute the output. Truth tables are foundational in logic, computer science, and circuit design. They make abstract logical relationships concrete and visualizable.
Basic operators. AND (∧): true only if BOTH parts true. T-T → T; otherwise F. OR (∨): true if EITHER part true. F-F → F; otherwise T. NOT (¬): flips truth. T → F; F → T. IF...THEN (→, conditional): false only when antecedent is true and consequent false. T-F → F; otherwise T. IFF (↔, biconditional): true when both have same truth value. The truth table for any compound statement is built from these basic operators.
For "P AND Q," when is the statement TRUE?
Practical use. Computer LOGIC GATES (AND, OR, NOT, NAND, NOR, XOR) implement truth tables in hardware. Combinations build everything from adders to CPUs. Truth tables also expose tricky cases. The CONDITIONAL "if P then Q" being false ONLY when P is true and Q is false leads to surprises (e.g., a vacuously-true statement with false antecedent). Logical equivalences (DeMorgan's laws, etc.) can be verified with truth tables.
Build One
Build a truth table for "(P AND Q) OR (NOT P)." Enumerate all 4 combinations of T/F for P and Q. Compute step by step. Notice the pattern — it's actually equivalent to "Q OR (NOT P)." Working through truth tables sharpens logical thinking.
Truth tables make logic mechanical and visualizable. They underlie computer hardware, mathematical proofs, and clear thinking.
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