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Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
Question
Let f: X $\to$ Y is 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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Answer

The given function is f : X $\to$ Y and relation on X is R = {(a, b) : f (a) = f (b)}
Reflexive Since, for every x$\in$X, we have
$f(x) = f(x)$
$\Rightarrow$ $(x, x)$$\in$R $\forall$ x$\in$X
$\therefore$ R is reflexive.

Symmetric Let (x, y)$\in$R
Then, $f(x)=f(y)$
$\Rightarrow f(y) = f(x) $
$\Rightarrow (y, x)$$\in$R
$\therefore$ R is symmetric.

Transitive Let x, y, z$\in$X such that (x, y)$\in$Rand (y, z)$\in$R
Then $f(x) = f(y)$........(i)
and $f (y) = f (z)$.......(ii)
From Equation (i) and (ii), we get
$\Rightarrow$$f(x) = f(z)$
$\Rightarrow$ (x, z) $\in$ R
Thus, (x, y)$\in$R and (y, z)$\in$R
$\Rightarrow$ (x, z) $\in$ R $\forall$ x, y, z$\in$X
$\therefore$ R is transitive.
Therefore, R is transitive. Since, R is reflexive, symmetric and transitive, so it is an equivalence relation.

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