Question

Let X = {1,2,3,4,5,6,7,8,9). Let Ry be a relation on X given by R1 = {(x, y): x - y
is divisible by 3) and R2 be another relation on X given by R2 = {(x, y): {x, y} c{1,4,7} or
{x, y} = {2,5, 8}or {x,y} c{3,6,9}}. Show that Ry R2

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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`

Answer 2

Answer:

Using Venn diagram it is very clear that A−B=A∩B

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