Q. 14 D5.0( 1 Vote )

# 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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Fill in the blanks in each of the

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.

Mathematics - ExemplarFill in the blanks in each of the

Let the relation R be defined on the set

A = {1, 2, 3, 4, 5} by R = {(a, b) : |a^{2} – b^{2}| < 8}. Then R is given by _______.

State True or False for the statements

Every relation which is symmetric and transitive is also reflexive.

Mathematics - ExemplarState True or False for the statements

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

Mathematics - Exemplar