Q. 3 B4.0( 5 Votes )

# Test whether the following relations R1, R2 and R3 are (i) reflexive (ii) symmetric and (iii) transitive :R2 on Z defined by (a, b) ϵ R2⇔ |a – b| ≤ 5

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

R2 on Z defined by (a, b) R2 |a – b| ≤ 5

Check for Reflexivity:

a Z,

(a, a) R2 needs to be proved for reflexivity.

If (a, b) R2

Then, |a – b| ≤ 5 …(1)

So, for (a, a) R1

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

|a – a| ≤ 5

0 ≤ 5

(a, a) R2

So, a Z, then (a, a) R2

R2 is reflexive.

Check for Symmetry:

a, b Z

If (a, b) R2

We have, |a – b| ≤ 5 …(2)

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

|b – a| ≤ 5

Since, the value is in mod, |b – a| = |a – b|

The statement |b – a| ≤ 5 is true.

(b, a) R2

So, if (a, b) R2, then (b, a) R2

a, b Q0

R1 is symmetric.

Check for Transitivity:

a, b, c Z

If (a, b) R2 and (b, c) R2

|a – b| ≤ 5 and |b – c| ≤ 5

Since, inequalities cannot be added or subtract. We need to take example to check for,

|a – c| ≤ 5

Take values a = 18, b = 14 and c = 10

Check: |a – b| ≤ 5

|18 – 14| ≤ 5

|4| ≤ 5 is true.

Check: |b – c| ≤ 5

|14 – 10| ≤ 5

|4| ≤ 5

Check: |a – c| ≤ 5

|18 – 10| ≤ 5

|8| ≤ 5 is not true.

(a, c) R2

So, if (a, b) R2 and (b, c) R2, then (a, c) R1

a, b, c Z

R2 is not transitive.

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