Q. 14 E5.0( 2 Votes )

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

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 which is transitive but neither reflexive nor symmetric.

Let there be a set A.

A = {1, 2, 3}

Transitive Relation:

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

This is neither reflexive nor symmetric.

(1, 1) R

(2, 2) R

(4, 4) R

Hence, R is not reflexive.

if (2, 4) R

Then, (4, 2) R

Hence, R is not symmetric.

Thus, the relation which is transitive but neither reflexive nor symmetric is:

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

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