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

Answer :

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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