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Let A, B and C be the sets such that A u...

Let `A`, `B` and `C` be the sets such that `A uu B=A uu C` and `A nn B = A nn C`. Show that `B=C`

Text Solution

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(i) `because` Sets A, B and C are such that
`A cup B = A cup C`
or `(A cup B) cap B = (A cupC) cap B`
or `(A cap B) cup (B cap B) = (A cap B) cup (C cap B)` (From distributive law)
or `B = (A cap B) cup (C cap B)` ….(1)
[Since `(A cap B) cup (B cap B)=B`]
Again `(A cup B) = (A cup C)`
or `(A cup B) cap C = (A cup C) cap C`
or `(A cap C) cup (B cap C)= (A cap C) cup (C cap C)` [From distributive law)
or `(A cap C) cup (B cap C) = C` [Since `(A cap B) cup (C cap C)=C`]
then `C = (A cap C) cup (B cup C)`...(2)
`because` Given `A cap B = A capC` then, replace `(A cap C)` by `(A cap B)` in equation (2),
`C = (A cap B) cup (B cup C)`...(3)
Now comparing equation (1) and equation (3),
`B=C`.
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