Give an example of a relation which is reflexive and transitive but no

1.	Give an example of a relation which is reflexive and transitive but not 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 on A which is reflexive and transitive but not symmetric. Let there be a set A. A = {1, 2, 3, 4} Reflexive relation: R = {(1, 1), (2, 2), (3, 3), (4, 4)} …(1) Transitive relation: R = {(3, 4), (4, 1), (3, 1)} …(2) Combine results (1) and (2), we get R = {(1, 1), (2, 2), (3, 3), (4, 4), (3, 4), (4, 1), (3, 1)} Check for Symmetry: If (3, 4) ∈ R Then, (4, 3) ∉ R ∀ 3, 4 ∈ A [∵ A = {1, 2, 3, 4}] One example is enough to prove that R is not symmetric. Thus, the relation which is reflexive and transitive but not symmetric is: R = {(1, 1), (2, 2), (3, 3), (4, 4), (3, 4), (4, 1), (3, 1)}

Give an example of a relation which is reflexive and transitive but not 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 on A which is reflexive and transitive but not symmetric.

Let there be a set A.

A = {1, 2, 3, 4}

Reflexive relation:

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

Transitive relation:

R = {(3, 4), (4, 1), (3, 1)} …(2)

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

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

Check for Symmetry:

If (3, 4) ∈ R

Then, (4, 3) ∉ R

∀ 3, 4 ∈ A [∵ A = {1, 2, 3, 4}]

One example is enough to prove that R is not symmetric.

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

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