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A truth table is a mathematical table used in logic to compute the truth value of logical expressions based on the truth values of their constituent propositional variables. It lists all possible combinations of truth values for the variables and shows the resulting truth value for the entire expression for each combination. Truth tables are fundamental tools in propositional logic, computer science, and other fields where logical reasoning is applied.

The construction of a truth table involves identifying the atomic propositions, which are the simplest statements that can be either true or false. For each atomic proposition, two columns are created, one for "true" (often represented by T or 1) and one for "false" (often represented by F or 0). If there are n atomic propositions, there will be 2^n possible combinations of truth values, and thus 2^n rows in the table.

Logical connectives such as AND (conjunction, symbolized by ∧), OR (disjunction, symbolized by ∨), NOT (negation, symbolized by ¬), IMPLIES (conditional, symbolized by →), and IFF (biconditional, symbolized by ↔) are used to combine atomic propositions into more complex logical statements. The truth value of a compound statement is determined by applying the rules of these connectives to the truth values of its components. For example, the conjunction of two propositions P and Q (P ∧ Q) is true only when both P and Q are true.

Truth tables are used to verify logical equivalences, determine whether an argument is valid, and design digital logic circuits. They provide a systematic and exhaustive method for analyzing the logical behavior of statements and systems.