Subsets

If every element in belongs to , we call a subset of and write

If is a subset of and there exists an element in which does not belong to , we call a proper subset and write

That is, a proper subset asserts that the two sets cannot be equal.

• , and
• , but

Sometimes there is . but this could mean proper subset or just subset depending on the person

Using only , formalize:

1. Esmerelda is a duck

\text{CSStudents} \subseteq \text

\neg(\text{DaysOfNov} \subseteq \text{Holidays})

1. The empty set is a subset of every set:
2. Every set is a subset of itself:
3. Subset is transitive:
4. Intersection is a subset:
5. Union is a superset:

(see Set Equality)

A superset is just the opposite of a subset. That is, if every element of belongs to , we call a superset of and write .

Two sets are disjoint or mutually exclusive if and only if

In other words, they share no common elements, so their intersection is empty.

Given arbitrary sets , we say these sets are pairwise disjoint if all of them are disjoint with each other. That is, none of them share common elements.