Question

28. Let R: Z → Z be a relation defined by R= {(a, b) | a, b e Z, a-b e Z}
(i) V a e Z, (a, a) ER
(ii) (a, b) ER= (b, a) ER
(iii) (a, b), (b, c) ER=(a, c) ER​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{In Z}$

$\mathsf{_aR_b\;\iff\;a-b\;\in\;Z}$

$\underline{\textsf{To prove:}}$

$\textsf{R is an equivalance relation}$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{_aR_b\;\iff\;a-b\;\in\;Z}$

$\underline{\textsf{Reflexive:}}$

$\textsf{For}\;\mathsf{a\,\in\,Z}$

$\mathsf{a-a=0,}\;\textsf{which is an integer}$

$\mathsf{a-a\;\in\;Z}$

$\implies\mathsf{_aR_a}$

$\textsf{Hence R is reflexive}$

$\underline{\textsf{Symmetric:}}$

$\textsf{Let}\;\mathsf{_aR_b}$

$\implies\mathsf{a-b}\;\textsf{is an integer}$

$\implies\mathsf{b-a}\;\textsf{is also an integer}$

$\implies\mathsf{b-a\;\in\;Z}$

$\implies\mathsf{_bR_a}$

$\textsf{Hence R is symmetric}$

$\underline{\textsf{Transitive:}}$

$\textsf{Let}\;\mathsf{_aR_b}\;\textsf{and}\;\mathsf{_bR_c}$

$\implies\textsf{(a-b) and (b-c) are integers}$

$\textsf{We know that, Sum of two integers is again an integer}$

$\implies\mathsf{(a-b)+(b-c)}\;\textsf{is also an integer}$

$\implies\mathsf{a-c}\;\textsf{is also an integer}$

$\implies\mathsf{a-c\;\in\;Z}$

$\implies\mathsf{_aR_c}$

$\textsf{Hence R is transitive}$

$\therefore\textsf{R is an equivalence relation}$

Find more:

A relation R is defined on the set of positive integers as xRy if 2x + y = 5. Therelation Ris

(A) reflexive

(B) transitive

(C) symmetric

(D) not transitive

https://brainly.in/question/24108840