Question

Let X = {2, 4, 5, 6} Y = {3, 4, 7, 8} Z = {5, 6, 7, 8}

a) (X - Y) ∩ (Y - X)​

Answer 1

Given

• X = {2, 4, 5, 6}
• Y = {3, 4, 7, 8}
• Z = {5, 6, 7, 8}

To Find

• (X - Y) ∩ (Y - X)

Solution

☯ This → ∩ symbol represents the intersection between the two sets

━━━━━━━━━━━━━━━━━━━━━━━━━

✭ According to the Question :

➞ (X - Y) ∩ (Y - X)

• X = {2, 4, 5, 6}
• Y = {3, 4, 7, 8}
• Z = {5, 6, 7, 8}

➞ {2,5,6} ∩ {3,7,8}

• But here we may observe that there is no common terms on both the sides and hence there would be no intersection between the two sets

∴ (X - Y) ∩ (Y - X) = ∅

Answer 2

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`