Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.Solution:

Sure, I can do that. Here is the solution:

We are given that A ∪ B = A ∪ C and A ∩ B = A ∩ C. We need to prove that B = C.

Let's take an element x in B. Since A ∪ B = A ∪ C, x is also in A ∪ C. This means that x is in A or x is in C. If x is in A, then since A ∩ B = A ∩ C, x is also in C. If x is not in A, then x must be in C. Therefore, every element in B is also in C, so B is a subset of C.

Now let's take an element y in C. Since A ∪ B = A ∪ C, y is also in A ∪ B. This means that y is in A or y is in B. If y is in A, then since A ∩ B = A ∩ C, y is also in B. If y is not in A, then y must be in B. Therefore, every element in C is also in B, so C is a subset of B.

Since B is a subset of C and C is a subset of B, B = C.