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.

Algebra ->  sets and operations -> 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.       Log On


   

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) About Me  (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.