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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 ...
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).
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 negated and swapped.
Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle. Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.
definition: is defined as metalanguage:= means "from now on, is defined to be another name for ." This is a statement in the metalanguage, not the object language. The notation may occasionally be seen in physics, meaning the same as :=.
Going from a statement to its converse is the fallacy of affirming the consequent. However, if the statement S and its converse are equivalent (i.e., P is true if and only if Q is also true), then affirming the consequent will be valid. Converse implication is logically equivalent to the disjunction of and
For a third facet, identify every mathematical predicate N with the set T(N) of objects, events, or statements for which N holds true; then asserting the necessity of N for S is equivalent to claiming that T(N) is a superset of T(S), while asserting the sufficiency of S for N is equivalent to claiming that T(S) is a subset of T(N).
In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry ), a notion (for example, ellipse or minimal surface ) may have more than one definition.