Q. 9 A4.0( 5 Votes )

# If A = {1, 2, 3, 4}, define relations on A which have properties of beingreflexive, transitive but not 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.

Using these properties, we can define R on A.

A = {1, 2, 3, 4}

We need to define a relation (say, R) which is reflexive, transitive but not symmetric.

Let us try to form a small relation step by step.

The relation must be defined on A.

Reflexive relation:

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

Transitive relation:

R = {(1, 2), (2, 1), (1, 1)}, is transitive but also symmetric.

So, let us define another relation.

R = {(1, 3), (3, 2), (1, 2)}, is transitive and not symmetric. …(2)

Let us combine (i) and (ii) relation.

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

(A) can be shortened by eliminating (3, 2) and (1, 2) from R.

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

Further (B) can be shortened by eliminating (2, 2) and (4, 4).

R = {(1, 1), (3, 3), (1, 3)} …(C)

All the results (A), (B) and (C) is correct.

Thus, we have got the relation which is reflexive, transitive but not symmetric.

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