Mathematics and Statistics · Ch 1 — Mathematical Logic
Truth Tables of Compound Statements
Truth Tables of Compound Statements
The truth value of a compound statement is fixed entirely by the truth values of its simple parts and the connectives joining them. A truth table lists, row by row, every possible combination of truth values of the simple statements together with the resulting truth value of the compound statement. If a compound statement involves distinct simple statements, its truth table has rows — rows for one statement, for two, for three, and so on.
The five basic tables. Every larger table is built from these.
Negation :
| T | F |
| F | T |
Conjunction — true only when both are true:
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction — false only when both are false:
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Conditional — false only when the antecedent is true and the consequent false:
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The two rows where is false may look surprising: when the hypothesis is false, the conditional is counted true by convention ("vacuously true") — a promise "if it rains, I will carry an umbrella" is not broken on a day it does not rain.
Biconditional — true exactly when both parts have the same truth value:
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
A table listing every possible combination of truth values of the simple statements in a compound statement, with the resulting truth value of the compo …
A compound statement built from n distinct simple statements has a truth table with 2^n rows, one for each combination of T/F valu …