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If A, B and C be sets. Then, show that A...

If A, B and C be sets. Then, show that `A nn (B uu C) = (A nn B) uu (A nn C)`.

Text Solution

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Let `x in A nn (B uu C)`
`rArr x in A` and `x in (B uu C)`
`rArr x in A` and (`x in B` or `x in C`)
`rArr` (`x in A` and `x in B`) or `(x in A "and" x in C)`
`rArr x in A nn B` or `x in A nn C`
`x in (A nn B) uu (A nn C)`
`rArr A nn (B uu C) sub (A nn B) uu (A nn C)`
`rArr A nn (B uu C) sub (A nn B) uu (A nn C)`
Again, let `y in (A nn B) uu (A nn C) "......"(i)`
`rArr y in (A nn B)` or `y in (A nn C)`
`rArr (y in A "and" y in B)` or `(y in A "and" y in C)`
`rArr y in A` and `(y in B "or" y in C)`
`rArr y in A` and ` y in B uu C`
`rArr y in A nn (B uu C)`
`rArr (A nn B) uu (A nn C) sub A nn (B uu C)"......."(ii)`
From Eqs. (i) and (ii).
`A nn (B uu C) = (A nn B) uu (A nn C)`
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