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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. [1]
The white area shows where the statement is false. Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P). In general, the truth of S says nothing about the truth of its converse, [2] unless the antecedent P and the consequent Q are logically equivalent.
is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence , depends on the author’s style. x + 5 = y + 2 ⇔ x + 3 = y {\displaystyle x+5=y+2\Leftrightarrow x+3=y}
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
Some of these connectives may be defined in terms of others: for instance, implication, p → q, may be defined in terms of disjunction and negation, as ¬p ∨ q; [71] and disjunction may be defined in terms of negation and conjunction, as ¬(¬p ∧ ¬q). [48]
The rule states that P implies Q is logically equivalent to not-or and that either form can replace the other in logical proofs. In other words, if P {\displaystyle P} is true, then Q {\displaystyle Q} must also be true, while if Q {\displaystyle Q} is not true, then P {\displaystyle P} cannot be true either; additionally, when P {\displaystyle ...
Logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgan's laws: ¬(p ∧ q) ⇔ ¬ p ∨ ¬ q ¬(p ∨ q) ⇔ ¬ p ∧ ¬ q. Propositional variables become variables in the Boolean ...
It may be defined either by appending one of the two equivalent axioms (¬q → p) → (((p → q) → p) → p) or equivalently p∨(¬q)∨(p → q) to the axioms of intuitionistic logic, or by explicit truth tables for its operations. In particular, conjunction and disjunction are the same as for Kleene's and Łukasiewicz's logic, while the ...