Question

If A U B = A U C, then does this imply B=C?
Prove it with proper justification.
Please don't write if you don't know the answer.​

Answer 1

Answer:

It is actually not so hard so show a slightly stronger result, namely that it follows that A=B . So let us do that here:

So we assume that:

(1) A∪B=A∪C

(2) A∩B=A∩C

From (1) we can derive that (with A¯¯¯¯ denoting the complement of A ):

(3) ( A∪B)∩A¯¯¯¯=(A∪C)∩A¯¯¯¯

Distribution of ∩ over ∪ gives:

(4) (A∩A¯¯¯¯)∪(B∩A¯¯¯¯)=(A∩A¯¯¯¯)∪(C∩A¯¯¯¯)

Since A∩A¯¯¯¯=∅ , and X∪∅=X , we get:

(5) B∩A¯¯¯¯=C∩A¯¯¯¯

Note that at this stage we know that B and C are equal inside A , because of (2), and are also equal outside of A , because of (5). Formally, this allows us to show the following (here U is the universal set of which all sets are a subset):

B=

using X=X∩U

B∩U=

using U=X∪X¯¯¯¯

B∩(A∪A¯¯¯¯)=

using distribution of ∩ over ∪

(B∩A)∪(B∩A¯¯¯¯)=

using (2) and (5)

(C∩A)∪(C∩A¯¯¯¯)=

using distribution of ∩ over ∪

C∩(A∪A¯¯¯¯)=

using U=X∪X¯¯¯¯

C∩U=

using X=X∩U

C