Question

Suppose \(A, B\) and \(C\) are sets. Prove that if \(A \subseteq B\), then \(A-C \subseteq B-C\).

Short answer

The proof shows that if \(A \subseteq B\), then the set difference \(A-C\) is a subset of \(B-C\). The key element to this proof is the understanding of subset relationships and set operations, particularly set differences.

Step-by-step solution

Understand the Set Difference

The difference between two sets, denoted as \(A-C\), means all the elements in set A that are not in set C. For this exercise, we are subtracting C from both A and B. And we need to prove that if set A is a subset of set B, then the difference between A and C is also a subset of the difference between B and C.

Proof by Elements

To prove this, let's consider an arbitrary element \(x\) in \(A - C\). By the definition of set subtraction, this means that \(x\) is in A but not in C.

Apply Given Condition

Because we know from the given condition that if an element is in A, then it must also be in B (since \(A \subseteq B\)), therefore, \(x\) is in B.

Complete the Proof

Since \(x\) is in B and not in C, we can safely say that, \(x\) is also in the set \(B-C\). Hence, we prove \(A-C \subseteq B-C\) for all elements in \(A - C\), which will then apply to the entire set.