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.