Q. 3 A4.2( 10 Votes )

# Test whether the following relations R1, R2 and R3 are (i) reflexive (ii) symmetric and (iii) transitive :R1 on Q0 defined by (a, b) ϵ R1⇔ a = 1/b

Here, R1, R2, R3, and R4 are the binary relations.

So, 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.

So, using these results let us start determining given relations.

We have

R1 on Q0 defined by (a, b) R1

Check for Reflexivity:

a, b Q0,

(a, a), (b, b) R1 needs to be proved for reflexivity.

If (a, b) R1

Then, …(1)

So, for (a, a) R1

Replace b by a in equation (1), we get

But, we know

(a, a) R1

So, a Q0, then (a, a) R1

R1 is not reflexive.

Check for Symmetry:

If (a, b) R1

Then, (b, a) R1

a, b Q0

If (a, b) R1

We have, …(2)

Now, for (b, a) R1

Replace a by b & b by a in equation (2), we get

(b, a) R2

So, if (a, b) R1, then (b, a) R1

a, b Q0

R1 is symmetric.

Check for Transitivity:

If (a, b) R1 and (b, c) R1

We need to eliminate b.

We have

Putting in , we get

But,

(a, c) R1

So, if (a, b) R1 and (b, c) R1, then (a, c) R1

a, b, c Q0

R1 is not transitive.

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