Mathematical Notation

Mathematical notation is dense for a reason: it compresses long English sentences into symbols you can scan in a second. "The sum, as $k$ ranges over the integers from 1 to 100, of $k$ squared" becomes $\sum_{k=1}^{100} k^2$ — same meaning, fraction of the space. This section is a tour of the symbols you will need to fluently read every later chapter, and every math text you ever open.

The connectives and quantifiers

$\in$ — "is an element of." $3 \in \mathbb{Z}$.

$\notin$ — "is not an element of."

$\subseteq$ — "is a subset of."

$\Rightarrow$ — "implies." $P \Rightarrow Q$ is the conditional statement from §Int.2.

$\Leftrightarrow$ — "if and only if." True exactly when $P$ and $Q$ have the same truth value.

$\forall$ — the universal quantifier, "for all." $\forall x \in \mathbb{R}, x^2 \geq 0$ reads "for every real $x$, $x^2$ is non-negative."

$\exists$ — the existential quantifier, "there exists." $\exists x \in \mathbb{R}, x^2 = 2$ reads "there exists a real $x$ with $x^2 = 2$" (true; $x = \sqrt{2}$).

The reference panel below catalogues these and other symbols. Click any one to see its name, how to say it aloud, and one concrete sentence using it — the spoken form is the part that makes reading mathematics out loud feel natural.

Summation and product

The summation symbol packages a sum of indexed terms:

The product symbol packages a product:

The factorial is a special product: $n! = \prod_{i=1}^{n} i = 1 \cdot 2 \cdot 3 \cdots n$. By convention $0! = 1$ (this convention makes formulas like $\binom{n}{0} = 1$ work without exception).

Reading and writing quantifier statements

The order of quantifiers MATTERS. $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}, y > x$ is true (for every real, there is a larger real). But $\exists y \in \mathbb{R}, \forall x \in \mathbb{R}, y > x$ is false (it claims a single $y$ greater than everyone, which would have to be larger than itself).

The negation rules: $\neg(\forall x, P(x)) = \exists x, \neg P(x)$ and $\neg(\exists x, P(x)) = \forall x, \neg P(x)$. To negate a universal claim, find a single counterexample; to negate an existential claim, show none exists.

Try it

• Evaluate $\sum_{k=1}^{4} k^2$. Then evaluate $\sum_{k=1}^{4} (2k - 1)$. Compare the two.
• Compute $\prod_{i=1}^{4}(i+1)$. Express it as a factorial.
• Translate to symbols: "there exists an integer whose square is 49." Now translate: "for every positive integer $n$, the factorial $n! \geq 1$."
• What is the negation of "$\forall x \in \mathbb{Q}, x \neq 0$"? Is the original statement true?

A trap to watch for

Swapping quantifier order silently changes the meaning. "Every student in this class has a favourite number" ($\forall s, \exists n$) does NOT say "there is a number that every student in this class favours" ($\exists n, \forall s$). The first is almost certainly true; the second almost certainly false. The fix: when reading $\forall \ldots \exists \ldots$, mentally unfold it into English with the "different choice for each" pattern; for $\exists \ldots \forall \ldots$, use the "one choice that works for everyone" pattern.

What you now know

You can read and write summation, product, quantifier, and set-membership notation fluently. This is the last Intermezzo section — the next chapter returns to mathematics proper, opening with the geometry of distance and angles on the plane.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Intermezzo, §4 — mathematical notation, summation, and quantifiers.
• Velleman, D. (2019). How to Prove It: A Structured Approach, 3rd ed. Cambridge. Chapter 2 — quantifiers and their negations.
• Houston, K. (2009). How to Think Like a Mathematician. Cambridge. Chapter 5 — reading and writing mathematical notation deliberately.