Sets and Elements

Chapter 2: Reading, Writing, and Notating Mathematics

Learning objectives

Define sets using roster and set-builder notation
Understand subset, union, and intersection
Work with the empty set

Modern mathematics speaks set theory the way physics speaks calculus. Every mathematical object — a number, a function, a vector, a triangle — lives in some set. Every theorem is a claim about elements of a set. Get fluent here and the rest of mathematics opens up. A set is a well-defined collection of objects called elements (or members). We write $a \in A$ to mean " $a$ is an element of $A$ " and $a \notin A$ to mean " $a$ is not in $A$ ."

Set Notation

Roster: ${1, 2, 3}$ lists all elements.

Set-builder: ${x \in \mathbb{Z} : x > 0}$ describes elements by a property.

Set Operations

Union:

A \cup B = \{x : x \in A \text{ or } x \in B\}

Intersection:

A \cap B = \{x : x \in A \text{ and } x \in B\}

Subset: $A \subseteq B$ means every element of $A$ is also in $B$ .

Empty set: $\emptyset = {}$ has no elements. It is a subset of every set.

Where this shows up

- **Databases:** SQL operations — INTERSECT, UNION, EXCEPT — are literally set intersection, union, and difference; the empty set corresponds to a zero-row result. - **Probability:** Events in probability are subsets of a sample space, and probabilities of

A \cup B

and

A \cap B

follow the inclusion-exclusion law, which is set theory dressed up in numbers. - **Type Systems:** Type-checkers in languages like TypeScript treat each type as a set of possible values; subtyping is set inclusion, union types are unions, and the never type is the empty set.

(Click a dot to move it between regions of the Venn diagram. Watch the readout update with |A|, |B|, |A ∪ B|, and |A ∩ B|. Toggle to 3-set mode for a more complex picture.)

Two famous identities

The inclusion-exclusion principle for two sets: $|A \cup B| = |A| + |B| - |A \cap B|$ . The intersection is subtracted because elements in both sets would otherwise be double-counted.

The distributive law: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ . This mirrors $a(b + c) = ab + ac$ from arithmetic, with $\cap$ playing the role of multiplication and $\cup$ the role of addition.

Try it

In the widget, place dots so that $|A| = 3, |B| = 3, |A \cap B| = 2$ . Read off $|A \cup B|$ . Confirm it equals $3 + 3 - 2 = 4$ .
Let $A = {1,2,3,4}$ and $B = {3,4,5,6}$ . Find $A \cup B$ and $A \cap B$ .
List the elements of ${x \in \mathbb{Z} : x^2 < 10}$ . (Hint: check the integers between $-3$ and $3$ .)
Switch the widget to 3-set mode. Place a dot in the central A ∩ B ∩ C region. Now move it to the A ∩ B but NOT C region. What changes in the summary?

A trap to watch for

Beginners write ${a, b} = {a, a, b}$ and worry. In set theory, elements are not repeated — either an object is in the set or it is not, and listing it twice does not change the set. Also, order does not matter: ${a, b} = {b, a}$ . If you need order, you want an ordered pair or tuple, not a set. The fix: when a set looks like it has duplicates, mentally collapse them; when order matters, use parentheses $(a, b)$ instead of braces.

What you now know

You can describe sets in roster or set-builder form, perform union, intersection, and subset operations, and articulate the role of the empty set. The next section catalogues the symbolic notation that builds on these set foundations.

Quick check

References

Lang, S. (1971). Basic Mathematics. Springer. Intermezzo, §3 — sets, elements, and basic operations.
Halmos, P. (1974). Naive Set Theory. Springer. The classic gentle introduction to set theory.
Velleman, D. (2019). How to Prove It: A Structured Approach, 3rd ed. Cambridge. Chapter 2 — set theory presented alongside the proof techniques that use it.