Proof for sets of unions, need checking

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lookingforhelp

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For all integers n>=1, ifi A1, A2, A3,.... and B are any sets, then
U(Ai - B) = U(Ai) - B ,
where U is U from i=1 to n like below
n
U
i=1

My attempt: (using U as the U from i=1 to n, like above)
If x is an element of U(Ai-B), then x is an element of Ai - B for some i=1,2,...n and so
1) for some i=1,2,...n, x is an element of Ai and
2) x is not an element of B
Conversely, if x is an element of (U Ai) -B, then x is an element of U Ai, and x is not an element of B.
By definition of a general union, x is an element of Ai for some i=1,2,...,n, x is an element Ai and x is not an element of B.
So there must be an integer i such that x is an element of Ai - B, and thus that x is an element of U (Ai - B)

Is this correct, am I missing anything? Thanks for the help.
 
I agree with your proof. Just some comments.


lookingforhelp said:
For all integers n>=1, ifi A1, A2, A3,.... and B are any sets, then
U(Ai - B) = U(Ai) - B ,
where U is U from i=1 to n like below
n
U
i=1

My attempt: (using U as the U from i=1 to n, like above)
If x is an element of U(Ai-B), then x is an element of Ai - B for some i=1,2,...n and so
1) for some i=1,2,...n, x is an element of Ai and
2) x is not an element of B
Click to expand...


You didn't conclude anything. All proofs should conclude (concisely) with the statement you're after.

Conversely, if x is an element of (U Ai) -B, then x is an element of U Ai, and x is not an element of B.
By definition of a general union, x is an element of Ai for some i=1,2,...,n, x is an element Ai and x is not an element of B.
So there must be an integer i such that x is an element of Ai - B, and thus that x is an element of U (Ai - B)

Is this correct, am I missing anything? Thanks for the help.
Click to expand...

Second part is alright.
 
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