Every operator and what it does

One page for every symbol in this course that combines, compares, or transforms things.

Set operations

Symbol

$\in$

element-of (∈)

Is a member of.

Symbol

$\cup$

union (∪)

Union — everything in either set.

Symbol

$\cap$

intersection (∩)

Intersection — only what's in both.

Symbol

$\subset$

subset (⊂)

Is entirely contained in.

Symbol

$\varnothing$

empty set (∅)

The empty set.

Term

union

See section →

Symbol

\cup

— everything in either set, combined.

Term

subset

See section →

Symbol

\subset

— one set sits entirely inside another.

Term

disjoint

See section →

Two sets that share no elements — their intersection is empty.

Two sets sharing no elements at all.

Arithmetic and comparison

Symbol

$\times$

cross product (×, between two vectors)

Multiplication.

Symbol

$\div$

division (÷, the obelus)

Division (the obelus).

Symbol

$\pm$

plus-or-minus (±)

Plus-or-minus — both branches of a solution at once.

Symbol

$\le$

less-than-or-equal (≤)

Less than or equal.

Symbol

$\ge$

greater-than-or-equal (≥)

Greater than or equal.

Symbol

$\ne$

not-equal (≠)

Not equal.

Symbol

$\sqrt$

square/nth root (√)

Square or nth root.

Term

order of operations

See section →

The agreed sequence for evaluating an expression — parentheses, exponents, multiply/divide, add/subtract.

The referee when several operators collide in one expression.

Term

reciprocal

See section →

The number you multiply by to get 1 — flip the fraction.

Formula

Laws of Exponents

Exponents and radicals →

For a real number $a$ and positive integers $m, n$ :

a^n = \underbrace{a \cdot a \cdots a}_{n \text{ factors}}

The laws of exponents:

Product rule: $a^m \cdot a^n = a^{m+n}$
Quotient rule: $\dfrac{a^m}{a^n} = a^{m-n}$ (for $a \neq 0$ )
Power rule: $(a^m)^n = a^{mn}$
Power of a product: $(ab)^n = a^n b^n$
Power of a quotient: $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ (for $b \neq 0$ )

To make the quotient rule consistent for all exponents we define, for $a \neq 0$ : $a^0 = 1 \qquad\text{and}\qquad a^{-n} = \frac{1}{a^n}.$ With these definitions, all five laws hold for every integer exponent.

Formula

Properties of Radicals

Exponents and radicals →

Valid whenever both sides are defined:

Product property: $\sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}$
Quotient property: $\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$
Rational-exponent link: $\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} = a^{m/n}$

Function and vector operations

Symbol

$\circ$

degree/composition symbol (∘, °)

Function composition (or a degree mark, depending on context).

Term

composite function

See section →

A new function built by feeding one function's output into another —

(f\circ g)(x)=f(g(x))

Formula

Function Composition

Composition and inverse functions →

(f \circ g)(x) = f(g(x)),

apply $g$ first, then apply $f$ to the result. Its domain is every $x$ in the domain of $g$ for which $g(x)$ lies in the domain of $f$ .

Term

dot product

See section →

Multiply matching components and add — turns two vectors into one number measuring how aligned they are.

Theorem

Algebraic and Geometric Dot Product Agree

Dot product and projection →

For vectors $\mathbf{u}, \mathbf{v}$ with angle $\theta$ between them,

\mathbf{u}\cdot\mathbf{v} = u_1v_1+u_2v_2 = |\mathbf{u}|\,|\mathbf{v}|\cos\theta.

Show proof

Draw $\mathbf{u}$ and $\mathbf{v}$ from a common starting point $O$ , with tips at points $U$ and $V$ . The three points $O$ , $U$ , $V$ form a triangle: two sides have lengths $|\mathbf{u}|$ and $|\mathbf{v}|$ , meeting at angle $\theta$ ; the third side, from $U$ to $V$ , has length $|\mathbf{u}-\mathbf{v}|$ . The Law of Cosines applied to this triangle gives

|\mathbf{u}-\mathbf{v}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2|\mathbf{u}|\,|\mathbf{v}|\cos\theta

Now expand the left side directly from components. Writing $\mathbf{u}=\langle u_1,u_2\rangle$ and $\mathbf{v}=\langle v_1,v_2\rangle$ , so $\mathbf{u}-\mathbf{v} = \langle u_1-v_1,\ u_2-v_2\rangle$ :

|\mathbf{u}-\mathbf{v}|^2 = (u_1-v_1)^2 + (u_2-v_2)^2 = u_1^2+u_2^2+v_1^2+v_2^2 - 2(u_1v_1+u_2v_2)

The first four terms are exactly $|\mathbf{u}|^2+|\mathbf{v}|^2$ , and the last piece is $2(\mathbf{u}\cdot\mathbf{v})$ by the algebraic definition, so

|\mathbf{u}-\mathbf{v}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 - 2(\mathbf{u}\cdot\mathbf{v})

Both boxed expressions equal $|\mathbf{u}-\mathbf{v}|^2$ , so set them equal to each other and cancel the shared $|\mathbf{u}|^2+|\mathbf{v}|^2$ from both sides: