Consider the equivalence relation ~ on R given by a ~ b if and only if [a] = [b]. You do not need to prove that this is an equivalence relation. Let S be the set of equivalence classes of R under ~. Prove that S is a countably infinite set.

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Q: for an equivalence relation R on A, both sets, the class of equivalence classes of R —A/R— is a set.

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Q: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…

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A: We know that

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A: partial order means reflexive and antisymmetric.

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Q: let R be an equivalence relation and let [x] be the equivalence class of x, prove x belong to [x]

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Q: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…

A: Consider the given information.

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Q: Use a particular counterexample to explain why R fails to be an equivalence relation is the…

A: We take counter example   9R10,10R11⇒9R  11

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A: We need to prove 1) Reflexive 2) symmetric 3) transitive.

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