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Truth tables are a systematic representation of all possible truth values of logical expressions or digital circuits. Each row of the table corresponds to a distinct assignment of truth values to the input variables, and the columns list the resulting values of sub-expressions or output functions. They are fundamental in propositional logic, where they provide a straightforward method for verifying tautologies, contradictions, and logical equivalence. In digital design, truth tables describe the behavior of combinational circuits such as AND, OR, XOR gates, as well as more complex components like multiplexers and arithmetic logic units. By juxtaposing input patterns with outputs, designers can deduce minimal Boolean expressions using Karnaugh maps or algebraic simplification. Truth tables also serve as the basis for truth-functional semantics in formal languages and are used to evaluate symbolic logic proofs, truth-value assignments in modal logic, and to construct truth functions for multi-valued logics. Their simplicity and exhaustive nature make them an indispensable tool for educators, engineers, and researchers in mathematics, computer science, and related fields.