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For all sets a,B and C show that (A-B)nn...

For all sets a,B and C show that `(A-B)nn(A-C)=A-(BuuC)`

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