Question

If \(A, B\) and \(C\) are sets, then \((A \cap B)-C=(A-C) \cap(B-C)\).

Short answer

The statement that \( (A \cap B) - C = (A - C) \cap (B - C) \) is true and thus verified.

Step-by-step solution

Understanding the Terms

Sets are collections of different objects, referred to as elements. The intersection (\( \cap \)) of two sets A and B, written as \( A \cap B \), is a set that contains all elements that are in both A and B. The difference (\( - \)) of two sets A and B, written as \( A - B \), is a set that contains all elements that are in A but not in B.

Starting with the Left-hand Side Expression

\( (A \cap B) - C \) is the set containing all elements that are in both A and B but not in C. Carry out the operations in brackets first (\( A \cap B \)), and then subtract the set C from this.

Moving to the Right-hand Side Expression

\( (A - C) \cap (B - C) \) is the intersection of two sets. The first set \( A - C \) contains all the elements that are in A and not in C. The second set \( B - C \) contains all the elements that are in B but not in C. The intersection of these two sets would thus contain all elements that are in both \( A - C \) and \( B - C \).

Compare Both Sides

After carrying out the operations on both sides, it can be seen that both expressions will give the same elements, thus proving that \( (A \cap B) - C = (A - C) \cap (B - C) \). The intersection (\( A \cap B \)) that does not contain C equals the intersection of the elements from A that are not in C, with the elements from B that are not in C.