Q. 14 A5.0( 3 Votes )

Give an example of a relation which isreflexive and symmetric but not transitive.

Recall that for any binary relation R on set A. We have,

R is reflexive if for all x A, xRx.

R is symmetric if for all x, y A, if xRy, then yRx.

R is transitive if for all x, y, z A, if xRy and yRz, then xRz.

Let there be a set A.

A = {1, 2, 3, 4}

We need to define a relation on A which is reflexive and symmetric but not transitive.

Let there be a set A.

A = {1, 2, 3, 4}

Reflexive relation:

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

Symmetric relation:

R = {(3, 4), (4, 3)} …(2)

Combine results (1) and (2), we get

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

Check for Transitivity:

If (3, 4) R and (4, 3) R

Then, (3, 3) R

3, 4 A [ A = {1, 2, 3, 4}]

So eliminate (3, 3) from R, we get

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

Check for Transitivity:

If (4, 3) R and (3, 4) R

Then, (4, 4) R

3, 4 A

So, eliminate (4, 4) from R, we get

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

Thus, the relation which is reflexive and symmetric but not transitive is:

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

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