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equivalence relation
This term is a technical cornerstone of set theory and abstract algebra. It describes a specific type of relationship that allows elements of a set to be grouped together based on a shared property, effectively treating different elements as identical within a certain context. The three required properties—reflexivity, symmetry, and transitivity—must all be satisfied for the relation to qualify.
In mathematical discourse, this is distinct from a general relation or a function. It is most frequently used when defining quotient sets or partitions, where the focus shifts from individual elements to the equivalence classes they inhabit.
Ý nghĩa
Ví dụ
An equivalence relation must be reflexive, symmetric, and transitive.
Is this specific binary relation actually an equivalence relation?
The professor explained how an equivalence relation partitions a set into disjoint classes.
Modular congruence is a classic example of an equivalence relation.
Modular congruence is a classic example of an equivalence relation.
I wonder if I can define an equivalence relation that simplifies this proof.
We need to verify that the given relation is an equivalence relation before proceeding.
The concept of an equivalence relation is fundamental to abstract algebra.
The concept of an equivalence relation is fundamental to abstract algebra.
Let us assume that R is an equivalence relation on the set S.
Does every partition of a set correspond to a unique equivalence relation?
Does every partition of a set correspond to a unique equivalence relation?