Subset Difference Proof

Prove that if $A$ is a subset of $B$, then $A - C$ is a subset of $B - C$.

The concept of showing that one set is a subset of another set is fundamental in set theory. This problem focuses on applying this idea to set differences. When proving statements about subsets and set differences, it is often useful to work with element-based definitions. Consider what it means for an element to belong to the difference of two sets, such as A - C. From here, analyze how membership in this difference also implies membership in the difference B - C, if A is a subset of B. It's a useful exercise in understanding how logical implications work in set contexts. The strategy of this proof involves taking an arbitrary element from A - C and showing that it must also belong to B - C. This relies on the foundational definition that A is a subset of B, meaning every element of A is also an element of B. While solving this problem, students will reinforce their understanding of logical reasoning involving universal statements and the careful use of definitions, particularly around operations with sets like difference and subset. More broadly, this exercise is an example of deductive reasoning, a critical skill in both discrete mathematics and computer science, where precise problem solving underpins the creation of algorithms and the verification of program correctness.