Give an example of a relation which is symmetric but neither reflexive

1.	Give an example of a relation which is symmetric but neither reflexive nor transitive.
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 symmetric but neither reflexive nor transitive. Let there be a set A. A = {1, 2, 3, 4} Symmetric Relation: {(1, 3), (3, 1)} This is neither reflexive nor transitive. ∵ (1, 1) ∉ R (3, 3) ∉ R Hence, R is not reflexive. ∵ (1, 3) ∈ R and (3, 1) ∈ R Then, (1, 1) ∉ R Hence, R is not transitive. Thus, the relation which is symmetric but neither nor transitive is: R = {(1, 3), (3, 1)}

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

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 symmetric but neither reflexive nor transitive.

Let there be a set A.

A = {1, 2, 3, 4}

Symmetric Relation:

{(1, 3), (3, 1)}

This is neither reflexive nor transitive.

∵ (1, 1) ∉ R

(3, 3) ∉ R

Hence, R is not reflexive.

∵ (1, 3) ∈ R and (3, 1) ∈ R

Then, (1, 1) ∉ R

Hence, R is not transitive.

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

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

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