Question

Show that relation "is equal to" in sets is an equivalence relation.

Answer 1

Answer:

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Answer 2

A relation R on a set A is explained to be an equivalence relation if and only if the relation R is instinctive, symmetric and transitive. The equivalence relation is a connection on the pair which is normally exemplified by the numeral “∼”.

• Reflexive: A consideration is explained to be reflexive, if (a, a) ∈ R, for each a ∈ A.

• Symmetric: A consideration is explained to be symmetric if (a, b) ∈ R, again (b, a) ∈ R.

• Transitive: An association is explained to be transitive if (a, b) ∈ R and (b, c) ∈ R, therefore (a, c) ∈ R.

• In phrases of equivalence association inscription, it is interpreted as follows:

• A binary relation ∼ on a pair A is explained to be an equivalence consideration, if and only if it is instinctive, symmetric and transitive.
• x ∼ x (Reflexivity)
• x ∼ y if and only if y ∼ x (Symmetry)
• If x∼y and y∼z, then x∼z (Transitivity)
• Equivalence associations can be clarified in phrases of the subsequent examples:

• The indication of ‘is equal to (=)’ on a pair of quantities; for illustration, 1/3 = 3/9.
• For a provided pair of triangles, the association of ‘is related to (~)’ and ‘is congruent to (≅)’ exhibits equivalence.
• For a provided pair of numbers, the association of ‘congruence modulo n (≡)’ exhibits equivalence.
• The picture and domain are similar under a process, which exhibits the association of equivalence.

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