Set Builder Notation

There are two different styles in writing sets in set builder notation.

Style 1

S = {x ∈ T : P(x)} or alternatively S = {x ∈ T | P(x)}.
Read as: x is in T such that P(x)is true.

P(x) is called the membership criteria.
P(x) is an example of an open sentence:
a mathematical sentence whose truth value depends on the value of x, e.g., \(x^2\) −1 = 0(only true when x = ± 1).

For an object n to be in S, both n ∈ T and P(n) must be true.
If n ∈ T or P(n) is false, then n ∉ S.

Example: {n ∈ \(\mathbb{N}\): n < 1000 ∧ 7|n}
The members are: 7, 14, 21, . . . , 994,
i.e., all positive integers less than 1000 that are divisible by 7.

Style 2

S = {f(x):x ∈ T, P(x)} or alternatively S = {f (x) | x ∈ T, P(x)}.
Read as: f (x) is in T such that x satisfies P(x).

T has been already defined, and P(x) is still called the membership criteria.

f(x) is an expression in x, e.g., \(x^2\) − 1.

Example

all points on a circle of radius 8 centred at the origin

Solution:
S = {(x, y) ∈ \(\mathbb{R}\) * \(\mathbb{R}\) : \(x^2\) + \(y^2\) = 64} or S = {(8cos\(\Theta\), 8sin\(\Theta\)) : \(\Theta\) ∈ \(\mathbb{R}\), 0 \(\le\) \(\Theta\) \(\le\) 2\(\pi\) }