Let f:X→Y be a function. Define a relation R in X given by R={(a,b):f(a)=f(b)}. Examine whether R is an equivalence relation or not.
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Step-by-Step Solution
Step 1: Check for Reflexivity
For a relation to be reflexive, every element must be related to itself. This means for any element a in set X, the pair (a, a) must be in R. Since f(a) is always equal to f(a), the condition f(a)=f(a) is always true for any a∈X.
Step 2: Conclusion on Reflexivity
Since f(a)=f(a) is always true, it implies that (a,a)∈R for all a∈X. Therefore, the relation R is reflexive.
Step 3: Check for Symmetry
For a relation to be symmetric, if (a, b) is in R, then (b, a) must also be in R. If (a,b)∈R, it means f(a)=f(b). Since equality is symmetric, f(b) must also be equal to f(a). This implies that (b,a)∈R.
Step 4: Conclusion on Symmetry
Since f(a)=f(b) implies f(b)=f(a), the relation R is symmetric.
Step 5: Check for Transitivity
For a relation to be transitive, if (a, b) is in R and (b, c) is in R, then (a, c) must also be in R. If (a,b)∈R, then f(a)=f(b). If (b,c)∈R, then f(b)=f(c). By the transitive property of equality, if f(a)=f(b) and f(b)=f(c), then f(a)=f(c). This implies that (a,c)∈R.
Step 6: Conclusion on Transitivity
Since f(a)=f(b) and f(b)=f(c) implies f(a)=f(c), the relation R is transitive.