Math Notation

A set of elements a, b, c and d. An ellipsis can be used to denote 'and so on'.

Set of four numbers

$$\{-2,0,3,9\}$$

Set of 0, 2, 4 and so on (positive even numbers)

$$\{0,2,4,…\}$$

A set of elements containing items x such that a ≤ x ≤ b. Can use '(' or ')' to exclude endpoint.

[$$\begin{eqnarray} [-2,2] &=& \{-2,-1,0,1,2\} \\ (-2,2] &=& \{-1,0,1,2\} \\ [-2,2) &=& \{-2,-1,0,1\} \\ (-2,2) &=& \{-1,0,1\} \end{eqnarray}$$]

The empty set, ie the set that contains no elements

$$\varnothing = \{\}$$

The set of natural numbers

$$\mathbb{N} = \{1,2,3,4,…\}$$

The set of integers

$$\mathbb{Z} = \{…,-2,-1,0,1,2,…\}$$

The set of rational numbers (fractions). The set of numbers in the form \(^p/_q\) where \(p,q \in \mathbb{Z}\) and \(q \ne 0\)

[$$\begin{eqnarray} \frac{3}{4} &\in& \mathbb{Q} \\ \frac{\sqrt{3}}{4} &\notin& \mathbb{Q} \end{eqnarray}$$]

The set of real numbers. The set of all numbers between -∞ (negative infinity) and ∞ (infinity).

[$$\begin{eqnarray} \frac{\sqrt{3}}{4} &\in& \mathbb{R} \\ \mathbb{Z} &\subset& \mathbb{R} \end{eqnarray}$$]

Value x is a member of the set S

$$2 \in \{0,2,4,6\}$$

Value x is not a member of the set S

$$5 \notin \{0,2,4,6\}$$

Set C is a subset of D. Every element of set C is also an element of set D.

$$\{2,4\} \subset \{0,2,4,6\}$$

Set C is not a subset of set D. Set C contains at least one element that is not an element of set D.

$$\{3,4\} \not\subset \{0,2,4,6\}$$

The union of sets C and D. The set of elements contained in set C or set D or both.

$$\{0,2,4\} \cup \{4,6\} = \{0,2,4,6\}$$

The intersection of sets C and D. The set of elements in set C that are also elements of set D.

$$\{2,4\} \cap \{4,6\} = \{4\}$$

Complement of Set C (all elements that are not in set C)

For all.

The square of x is greater than one for all x greater than 1

$$x^2 > 1 \quad \forall \quad x > 1$$

There exists.

There exists a real number x such that x + 1 = 5 (i.e. x = 4)

$$\exists \; x \in \mathbb{R} \text{ such that } x+1=5$$

There does not exist.

There is no integer value of x such that x squared is 2

$$\nexists \; x \in \mathbb{N} \text{ such that } x^2 = 2$$

Such that. Can also use 'st'.

The negative real numbers is the set of values x such that x is between negative infinity and zero

$$\mathbb{R}^- = \{x:-\infty < x < 0 \}$$

There exists a real number x such that x + 1 = 5 (i.e. x = 4)

$$\exists \; x \in \mathbb{R} \text{ st } x + 1 = 5$$

Implies.

Where x takes the value of -2 implies that x squared is 4 (but x squared being 4 does not imply x is -2)

$$x=-2 \; \Rightarrow \; x^2 = 4$$

Implies and is implied by (equivalent to). Can also use 'iff' to mean 'if and only if'.

Where x takes the value of 0 implies that x cubed is also 0 (and vice-versa)

$$x=0 \; \Leftrightarrow \; x^3 = 0$$

The value n is even if, and only if, dividing by 2 produces an integer value

$$n \text{ is even iff } \frac{n}{2} \in \mathbb{N}$$

Tends to (or approaches). Can add '+' or '-' if approached from above or below respectively.

1 divided by x approaches 0 as x approaches infinity

$$\frac{1}{x} \to 0 \; \text{ as } \; x \to \infty$$

1 divided by x approaches 0 from above as x approaches infinity

$$\frac{1}{x} \to 0^+ \; \text{ as } \; x \to \infty$$

1 divided by x approaches 0 from below as x approaches negative infinity

$$\frac{1}{x} \to 0^- \; \text{ as } \; x \to -\infty$$