Variables and Sets

What is a set, and why is it the right starting point for all of mathematics? A set is just a collection of objects, with one rule: any object is either in the set or it is not — no partial memberships, no duplicates. That apparent triviality is the foundation that lets us talk about everything we want to talk about: numbers, points, functions, propositions, even other sets. Once you can write down a set, you can quantify over it, build operations on it, and reason about it precisely. In this section we cover the notation (membership, set-builder, the standard number systems) and the all-important distinction between free and bound variables — a distinction that decides whether a formula is a statement at all.

Membership and set-builder notation

If $A$ is a set, we write $x \in A$ for "$x$ is an element of $A$" and $x \notin A$ for "$x$ is not an element of $A$." A small set can be listed in roster notation: $A = {2, 3, 5, 7}$. Larger or infinite sets are described by a defining property using set-builder notation:

Read aloud: "the set of all $x$ such that $P(x)$." The bar $\mid$ separates the dummy variable from the defining property. Often we restrict the dummy variable to a known set first: ${x \in \mathbb{Z} \mid x > 0}$ is the set of positive integers.

The standard number systems

Four number systems appear so often they get reserved blackboard-bold names:

$\mathbb{N} = {0, 1, 2, 3, \ldots}$ — the natural numbers. Some authors start at 1; Velleman starts at 0.

— the integers (German Zahlen).

$\mathbb{Q}$ — the rational numbers, ratios $a/b$ with $a, b \in \mathbb{Z}$ and $b \neq 0$.

$\mathbb{R}$ — the real numbers, every point on the number line.

They sit in a tidy chain of inclusions: $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$. Each inclusion is strict: $-1 \in \mathbb{Z}$ but $-1 \notin \mathbb{N}$; $\tfrac{1}{2} \in \mathbb{Q}$ but $\tfrac{1}{2} \notin \mathbb{Z}$; $\sqrt{2} \in \mathbb{R}$ but $\sqrt{2} \notin \mathbb{Q}$.

Free vs. bound variables

A variable in a formula is bound if it is captured by a quantifier or by a set-builder; otherwise it is free. In ${x \mid x > y}$ the dummy variable $x$ is bound (its scope is the entire set definition) while $y$ is free — the meaning of the expression depends on what value of $y$ you have in mind. Renaming a bound variable does not change the set: ${x \mid x > y}$ and ${t \mid t > y}$ are exactly the same set.

A formula with no free variables is a statement: it has a definite truth value (true or false). A formula with free variables is sometimes called a predicate or an open sentence — its truth value depends on what values you plug into the free slots. So $x^2 = 4$ is not a statement (its truth depends on $x$), but $\forall x \in \mathbb{R}, x^2 \geq 0$ is a statement (it is true, period).

Pause and think: In the formula $\exists y \in \mathbb{Z}, y > x$, which variable is free and which is bound? If you replaced $x$ with the number $5$, would the result be a statement? (Hint: bound variables are absorbed by the quantifier; only free variables show up in the truth-condition.)

Try it

• Before listing: predict how many elements are in ${x \in \mathbb{Z} \mid -3 \leq x \leq 4}$. Then write them all out and check your count.
• Classify each by free/bound: in ${n \in \mathbb{N} \mid n^2 < k}$, name the bound variable, the free variable, and decide whether the formula is a statement.
• Predict first: is $\sqrt{9}$ in $\mathbb{N}$? Is it in $\mathbb{Z}$? Is it in $\mathbb{Q}$? Now verify by computing $\sqrt{9}$.
• Translate into set-builder notation: "all integers whose absolute value is less than 10." Then translate "the set of rational numbers strictly between $0$ and $1$."
• Pick a value of $x$ for which $x^2 < 5$ is true and one for which it is false. The formula itself is not a statement — explain why.

A trap to watch for

A surprisingly common mistake is treating set-builder notation as if it described a list rather than a set. Sets do not record order or multiplicity: ${1, 2, 2, 3} = {1, 2, 3} = {3, 1, 2}$. So when you write ${x \in \mathbb{Z} \mid x^2 \leq 4}$, you describe ${-2, -1, 0, 1, 2}$ — not five separate items in a particular order, just five members of one set. If a problem cares about order or repetition, you want a sequence or a tuple, not a set; that distinction will matter once you start studying ordered pairs in Chapter 4.

What you now know

You can read and write set-builder notation, navigate the chain $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R}$, and tell whether a formula is a statement by spotting its free variables. The next section combines pairs of sets with operations — union, intersection, complement — that mirror the logical connectives $\vee, \wedge, \neg$ you learned in §1.1.

References

• Velleman, D. J. (2019). How to Prove It: A Structured Approach (3rd ed.). Cambridge University Press, ch. 1.
• Halmos, P. R. (1960). Naive Set Theory. Van Nostrand, §§1–2.
• Enderton, H. B. (1977). Elements of Set Theory. Academic Press, ch. 2.
• Kunen, K. (2009). The Foundations of Mathematics. College Publications, ch. 1.