Read symbols as sentences

A set collects objects

The set $A=\{1,2,3\}$ contains three numbers. Order and repetitions do not change a set: $\{3,2,1,1\}=A$ . Write $2\in A$ for “2 belongs to A,” and $4\notin A$ for “4 does not belong to A.”

The symbol $\subseteq$ means “is a subset of.” Every element of $\{1,2\}$ belongs to $A$ , so $\{1,2\}\subseteq A$ . The empty set $\varnothing$ contains no elements and is a subset of every set. An element and a subset are different types of object: $2\in A$ , but $\{2\}\subseteq A$ .

Notation	Read it	Example
$A\cup B$	A or B, including both	$\{1,2\}\cup\{2,3\}=\{1,2,3\}$
$A\cap B$	A and B	$\{1,2\}\cap\{2,3\}=\{2\}$
$A\setminus B$	In A but not in B	$\{1,2\}\setminus\{2,3\}=\{1\}$
$\{x\in\mathbb R:x>0\}$	Real x such that x is positive	All positive real numbers

Intervals describe real-number sets: $[0,1]$ includes both endpoints, $(0,1)$ excludes both, and $[0,1)$ includes only the left endpoint. Infinity is not a real endpoint; use parentheses next to $\infty$ .

Tiny symbols, big differences

The quantifier $\forall$ means for every. The quantifier $\exists$ means there exists at least one. Compare:

\forall x\in\mathbb R,\ x^2\ge0\qquad\text{and}\qquad\exists x\in\mathbb R,\ x^2=4.

The first statement is about every real input. The second only needs one example, such as 2. The symbol $\Rightarrow$ means “implies,” and $\Leftrightarrow$ means “if and only if,” an implication in both directions.

Follow the index

The expression $\sum_{i=1}^{n}x_i$ says: visit positions $1,2,\ldots,n$ and add the values stored there. The subscript on $x_i$ is an address in a list; it is not a power.

Explore it yourself

Unpack the shorthand

The index is a moving finger. It visits each term; the big symbol tells you how to combine them.

Preparing interactive controls…

Σi = 1 … 4 i = 10

1 + 2 + 3 + 4 = 10

Last index n4

At i = 1, the term i contributes 1. The index names a position, not an extra factor.

For a list $x_1=3,x_2=5,x_3=2$ :

\sum_{i=1}^{3}x_i=3+5+2=10,\qquad\prod_{i=1}^{3}x_i=3\cdot5\cdot2=30.

In the experiment, the special choice $x_i=i$ makes each value equal to its index. Other lists need not have that property. Renaming the index does not change a sum: $\sum_{i=1}^3x_i=\sum_{j=1}^3x_j$ .

Work through an average

If three quiz scores are $6,8,10$ , their mean is

\frac{1}{n}\sum_{i=1}^{n}x_i=\frac{6+8+10}{3}=8.

Read from the inside outward: collect the scores, add them, then divide by the count. Later, expected value will use a weighted version of this idea.

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