Give an example of a relation which is transitive but neither reflexiv

1.	Give an example of a relation which is transitive but neither reflexive nor symmetric.
Answer» 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)}

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

Answer»

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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