Set Theory: Membership and Operations

Collections, Taken Seriously

A set is a collection of distinct objects, its members. Everything follows from two decisions: a set is fixed entirely by which objects belong to it, the axiom of extensionality, and it has no order and no repetition. So $\{1, 2\}$, $\{2, 1\}$ and $\{1, 2, 2\}$ are all the same set.

Sets are specified by listing, $\{1, 2, 3\}$, or by a condition, $\{x : x \text{ is even}\}$. The empty set $\emptyset$ has no members and is unique, since any two memberless sets have the same members and are therefore identical.

Two relations are constantly confused. Membership $a \in A$ holds between an object and a set. Inclusion $A \subseteq B$ holds between two sets, when every member of $A$ is also a member of $B$. They are different relations, and $\{1\} \in \{\{1\}\}$ while $\{1\} \subseteq \{1, 2\}$.

The operations combine sets:

The parallel with logic is exact and not a coincidence: union behaves like disjunction, intersection like conjunction, complement like negation. De Morgan's laws hold in both, in the same form.

Counting gives the inclusion-exclusion principle:

$$|A \cup B| = |A| + |B| - |A \cap B|$$

The subtraction is the whole content: adding the two sizes counts the overlap twice, so it must be removed once. For 30 students taking logic, 25 taking ethics and 10 taking both, the total is $30 + 25 - 10 = 45$, not 55.

Common pitfall: confusing $\in$ with $\subseteq$. The empty set is a subset of every set — vacuously, since it has no member that could fail to belong — but it is a member only of sets that explicitly contain it. $\emptyset \subseteq \{1\}$ is true; $\emptyset \in \{1\}$ is false.