Question

Let `X={1,2,3,4,5,6,7,8,9}.` Let R be a relation in `X` given by `R_1={(x,y):x-y` is divisible by `3}` and `R_2` another on `X` given by `R={(x,y):(x,y)uu{1,4,7}} or {x,y} uu {2,5,8} or {x,y} uu {3,6,9}}` Show that `R_1=R_2.`

Answer 1

Here, `X = {1,2,3,4,5,6,7,8,9}`
`R_1 = {(x,y): x-y` is divisible by `3}`
As, `x-y` is divisible by `3`.
`x-y = 3n` where `n in N`
`=>x= y+3n`
`y=1, x = 4` when `n=1`
`y=1, x = 7` when `n=2`
`y=4, x = 7` when `n=1`
`y=4, x = 1` when `n=-1`
`y=7, x = 4` when `n=-1`
`y=7, x = 1` when `n=-2`
`:. (x,y) = {(1,4),(1,7),(4,7),(4,1),(7,4),(7,1)}`
`:.(x,y) sub {1,4,7}`
Similarly, ` (x,y) = {(2,5),(2,8),(5,8),(5,2),(8,5),(8,2)}`
`:.(x,y) sub {2,5,8}`
Similarly, ` (x,y) = {(3,6),(3,9),(6,9),(6,3),(9,6),(9,3)}`
`:.(x,y) sub {3,6,9}`
`:. R_1 = {(x,y):(x,y) sub {1,4,7}} or {(x,y):(x,y) sub {2,5,8}} or {(x,y):(x,y) sub {3,6,9}}`
We are given,
`:. R_2 = {(x,y):(x,y) sub {1,4,7}} or {(x,y):(x,y) sub {2,5,8}} or {(x,y):(x,y) sub {3,6,9}}`
`:. R_1 = R_2`