Prove that the relation of equality in the set of
integers is an equivalence relation.
Answers
Given, set of Integers.
∴ R ={(a,b): a,b∈Z }
For, the relation to be equivalence class it should be
(i) Reflexive ,ie, R ∈ (x, x)
(ii) Symmentric ie, if (x,y)∈ Then, (y,x)∈R.
(iii) Transistive ie, if (x,y) and (y,z) ∈R .Then, (x,z)∈R
Ist case : Reflexive
given (a,b)∈ R .Then a,b∈Z
(a, a)∈R .Because , a∈Z.
Hence, the relation is reflexive.
2nd case : Symmetric
given (a,b)∈R . Then, a,b∈Z
⇒ (b,a)∈R as b, a∈Z.
Thus, the relation is symmetric.
3rd Case: Transistivity.
given (a,b)∈R . Let (b,c)∈Z
⇒ a,b,c ∈Z
⇒ (a,c)∈R
Hence, the relation is transistive.
Since, it is reflexive, symmetric and transistive. The given relation (Set) is equivalence class.
Answer:
hey,
If f(1) = g(1), then g(1) = f(1), so R is symmetric. If f(1) = g(1) and g(1) = h(1), then f(1) = h(1), so R is transitive. R is reflexive, symmetric, and transitive, thus R is an equivalence relation.
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