Question

show that R is an equivaence relation R defined on the set `S={1,2,3,4,5}` given by `R={(a,b):|a-b|` is even } is an equivalence relation .

Answer 1

(i) `|a-a|=0,` which is even .SO a R a.
(ii) a R b`implies |a-b|` is even
`implies |-(a-b)|` is even
`implies ` b R a.
(iii) a R b,b R c `implies |a-b|is even and |b-c| is even
`implies (A-b)` is even
`implies (a-c) ` is even
` implies a R c.`