SOLUTION: Let A, B, and C be sets such that A union B = A union C and A intersect B = A intersect C. Show that B=C.

Question 467910: Let A, B, and C be sets such that A union B = A union C and A intersect B = A intersect C. Show that B=C.
Answer by moshiz08(60) (Show Source): You can put this solution on YOUR website!
To show that two sets are equal, we need to show that every element of B is also an element of C, and that every element of C is also an element of B. Suppose x is in B. Then there are two cases.

Case 1. x is also in A. Then since x is in both A and B, we know that x is in A intersect B. Since A intersect B = A intersect C, this means that x is in A intersect C, which implies that x is in C.

Case 2. x is NOT in A. Then since x is in B, we know that x is in A union B. Since A union B = A union C, this means that x is in A union C, i.e. that x is in A or C. Since x is not in A, this implies that x is in C.

We have shown that any element of B is also in C. Similarly, by splitting into 2 cases like above, you can show that any element of C is also in B. Therefore the sets B and C have the same elements so B = C.
RELATED QUESTIONS
if A,B and C are mutually exclusive events and p(A) =0.2,p(B)=0.3 p(C)=0.2find (a) p(A... (answered by greenestamps)

(A union B) intersect negation (A intersect... (answered by tommyt3rd)

Let n(C) represent the number of elements in the set C. If n(A intersect B) is 17 and... (answered by jim_thompson5910)

(A union B)=(A union C) & (A intersection B)=(A intersection C) then PROVE THAT B=C (answered by MathLover1)

LET UNION = {1,2,3....,10}, A= {1,3,5,7,9}, B= {1,2,3,4,5,6,} and c={2,4,6,7,8} find... (answered by Fombitz)

Someone Please help me! Given the following A={1,2,3} B={1,2,3,4,5} and C= {4,5,6,7}... (answered by MathLover1)

Find the union of the sets {a,b,c,d,e} and... (answered by stanbon)

if a=b-c, then b=a union... (answered by ikleyn)