Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) R if and only if max {x, y} {3, 4}. Then among the statements () : The number of elements in R is 18, and () : The relation R is symmetric but neither reflexive nor transitive.

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Solution

Q.
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) $\in$ R if and only if max {x, y} $\in$ {3, 4}. Then among the statements

( $S_{1}$ ) : The number of elements in R is 18, and

( $S_{2}$ ) : The relation R is symmetric but neither reflexive nor transitive. [2025]

1 both are true

2 only ( $S_{2}$ ) is true

3 both are false

4 only ( $S_{1}$ ) is true

Ans.
(2)

R = {(0, 3), (3, 0), (0, 4), (4, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)}

$∴$ Number of elements in R = 16

$∴$ $S_{1}$ is false.

Now, (0, 0), (1, 1), (2, 2), (5, 5) $\notin$ R $∴$ R is not reflexive.

Again, let (a, b) $\in$ R then (b, a) $\in$ R

As max {a, b} = max {b, a} $∴$ R is symmetric.

Now, R is not transitive as (0, 3), (3, 1) $\in$ R but (0, 1) $\notin$ R.

$∴$ $S_{2}$ is true.

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