9More about subsets

If X is a subset of Y, we also say that Y is a superset of X, and we write Y ⊃ X. This is the same fact written the other way around, just as “x is less than y” and “y is greater than x” say the same thing:

Definition 4 (Superset). Y ⊃ X means that X ⊂ Y. We then say that Y is a superset of X.

We have already noticed that every set is a subset of itself. Subsets also chain together, which is the next thing to prove.

Proposition 3 (transitivity of inclusion). For any sets X, Y and Z, if X ⊂ Y and Y ⊂ Z, then X ⊂ Z.

Figure 2. Proposition 3 (transitivity of inclusion): subsets chain together.

Let x be an arbitrary element of X. Since X ⊂ Y, we have x ∈ Y. Since Y ⊂ Z, we have x ∈ Z. Hence every element of X is an element of Z.

Sometimes we want to say that X is a subset of Y and that the two sets are not identical. In that case we say that X is a proper subset of Y, and we write X ⊊ Y:

Definition 5 (Proper subset). X ⊊ Y means that X ⊂ Y and X ≠ Y.

We will hardly ever need this.