The truth table associated with the material conditional if p then q (symbolized as p → q) is as follows: The truth table associated with the logical implication p implies q (symbolized as p ⇒ q, or more rarely Cpq) is as follows: Logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, which produces a value of false if the first operand is true and the second operand is false, and a value of true otherwise. Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q.

The truth table for p OR q (also written as p ∨ q, Apq, p || q, or p + q) is as follows: Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if at least one of its operands is true. It can also be said that if p, then p ∧ q is q, otherwise p ∧ q is p. For all other assignments of logical values to p and to q the conjunction p ∧ q is false. In ordinary language terms, if both p and q are true, then the conjunction p ∧ q is true. Clearly, for the Boolean functions, the output belongs to a binary set, i.e. A function f from A to F is a special relation, a subset of A×F, which simply means that f can be listed as a list of input-output pairs. Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values.Ī truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.Ī truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, A XOR B). A truth table is a mathematical table used in logic-specifically in connection with Boolean algebra, boolean functions, and propositional calculus-which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables.

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