1.5 Set Builder Notation

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 1.1 Set

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Set Builder Notation

Set builder notation uses propositions (statement with a truth value) to define sets:

A = {x : P (x)}

Reads as "set $A$ equals the set of all elements $x$ such that proposition $p (x)$ is true"

Some people like to use $|$ for the "such that". We will be avoiding this because it looks too similar to the norm symbol.

{x \underset{avoid this}{|} P (x)}

In set builder notation, the name of the general element does not matter.

{x : P (x)} = {y : P (y)} = {θ : P (θ)}

Example

Let's define

T = {x : x is an integer, and 1 \leq x \leq 1024}

Written using 1.2 Element Enumeration, we would write the elements of $T$ as:

T = {1, 2, 3, 4, \dots, 1024}

The statement $- 1 \in T$ would be a false statement. Note that $P (x)$ is the statement.

Therefore, $- 1 \notin T$ would be a true statement.