Give an example of a relation which is transitive but neither ref… - Vidyadip

Question

Give an example of a relation which is transitive but neither reflexive nor symmetric.

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Answer

Let a relation R is defined as:
R = {(a,b): a < b}
For any a $\in$ R, we have (a,a) $\notin$ R as a cannot be strictly less than itself.
In fact, a = a,
Therefore, R is not reflexive.
Now, (1,2) $\in$ R but 2 > 1
$\Rightarrow$ (2,1)) $\notin$ R.
$\Rightarrow$ R is not symmetric.
Now, let (a,b), (b,c) $\in$ R
$\Rightarrow$ a < b and b < c
$\Rightarrow$ a < c
$\Rightarrow$ (a,c) $\in$ R
$\Rightarrow$ R is transitive.
Therefore, relation R is transitive but not reflexive and symmetric.

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