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In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and the associated symbol ≅ {\displaystyle \cong } ) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects.
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 Boolean algebra, this means that the logical operation signified by "+" is not the same as the inclusive disjunction signified by "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the exclusive disjunction signified by "XOR" or "⊕". Naturally, these variations in ...
In logic, it is used with two different but related meanings.It can refer to the if and only if connective, also called material equivalence. [3] This is a binary operation whose value is true when its two arguments have the same value as each other. [4]
The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. A simpler example is equality. Any number a {\displaystyle a} is equal to itself (reflexive).
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
False equivalence is a common result when an anecdotal similarity is pointed out as equal, but the claim of equivalence does not bear scrutiny because the similarity is based on oversimplification or ignorance of additional factors. The pattern of the fallacy is often as such:
Logical equivalence, where two statements are logically equivalent if they have the same logical content; Material equivalence, a relationship where the truth of either one of the connected statements requires the truth of the other