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In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p q ...
For example, the sentence "'Snow is white' is true" becomes materially equivalent with the sentence "snow is white", i.e. 'snow is white' is true if and only if snow is white. Said again, a sentence of the form "A" is true if and only if A is true. The truth of more complex sentences is defined in terms of the components of the sentence:
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
Venn diagram of (true part in red) In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement "if and only if" (often abbreviated as "iff " [1]), where is known as the antecedent, and the consequent.
Some examples of convenient definitions drawn from the symbol set { ~, &, (, ) } and variables. Each definition is producing a logically equivalent formula that can be used for substitution or replacement. definition of a new variable: (c & d) = Df s; OR: ~(~a & ~b) = Df (a ∨ b) IMPLICATION: (~a ∨ b) = Df (a → b) XOR: (~a & b) ∨ (a & ~b ...
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. For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am a human," which is not necessarily true.