### 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\(\pi\) }