Determine whether the relation is reflexive, symmetric and transi… - Vidyadip

Question

Determine whether the relation is reflexive, symmetric and transitive:
Relation R in the set A of human beings in a town at a particular time given by
R = {(x, y) : x is father of y}

✓

Answer

It is given that R = {(x, y) : x is father of y}
$\Rightarrow$ (x, x) $\notin$ R as x cannot be the father of himself.
$\Rightarrow$ R is not reflexive.
Now, if (x, y) $\in$ R, then x is the father of y.
$\Rightarrow$ But y is not father of x.
$\Rightarrow$ (y, x) $\notin$ R
$\Rightarrow$ R is not symmetric.
Now, let (x, y), (y, z) $\in$ R
$\Rightarrow$ x is the father of y and y is the father of z.
$\Rightarrow$ x is not the father of z.
$\Rightarrow$ Indeed x is the grandfather of z.
$\Rightarrow$ (x, z) $\notin$ R
$\Rightarrow$ R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.

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