Prove that the relation of congruence is an equivalence relation.

Question
Congruence
asked 2020-11-08
Prove that the relation of congruence is an equivalence relation.

Answers (1)

2020-11-09
Step 1
The definition of a congruence depends on the type of algebraic structure under consideration.
So, for the question “Prove that the relation of congruence is an equivalence relation.” we are taking in general.
Let \(a \equiv b\) is denoted as a is congruent to b.
Step 2
Now,
Reflexive:
\(Since\ a\ \equiv a\forall a,\ the\ relation\ \equiv is\ reflexive\).
Symmetry:
\(Since\ a \equiv b\ then\ b \equiv\ a, the\ relation\ \equiv\ is\ symmetry\).
Transitivity:
\(Since\ a \equiv b\ and\ b \equiv c\ then\ a \equiv c\ ,\ the\ relation\ \equiv is\ transitive\).
Hence, the relation of congruence is an equivalence relation.
0

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