Here is a simple example of a proof about subsets:
Claim 1. The set of integer multiples of 6 between 0 and 8 is a subset of the set of integer multiples of 3 between 0 and 18.
Proof. We use the definition of subset to show that the set of integer multiples of 6 between 0 and 18 is a subset of the set of integer multiples of 3 between 0 and 18. Letting \(A\) be the integer multiples of 6 between 0 and 18, i.e., \(A = \{0,6,12,18\}\), and \(B\) be the integer multiples of 3 between 0 and 18, i.e., \(B = \{0,3,6,9,12,15,18\}\), we see by inspection that every element of \(A\) is also an element of \(B\). Therefore \(A \subseteq B\) and we have proved that the set of integer multiples of 6 between 0 and 18 is a subset of the set of integer multiples of 3 between 0 and 18. □
And here is a more complicated proof in which you can’t just check by inspection:
Claim 2. The set of all integer multiples of 6 is a subset of the set of all integer multiples of 3.
Proof. We use the definition of subset to show that the set of integer multiples of 6 is a subset of the set of integer multiples of 3. Suppose \(a\) is an integer multiple of 6, i.e., \(a = 6n\) for some integer \(n\). Noticing that \(6n = 3(2n)\), and that \(2n\) is an integer by closure under multiplication, we see that \(3(2n)\) is an integer multiple of 3. We have thus shown that any integer multiple of 6 is also an integer multiple of 3, and so the set of integer multiples of 6 is a subset of the set of integer multiples of 3. □