Sets

131. Sets

A set is a collection of distinct elements

Membership

If $𝑆$ is a set and $𝑥$ is an element of $𝑆$ :

𝑥 \in 𝑆

If $𝑆$ is a set and $𝑥$ is not an element of $𝑆$ :

𝑥 \notin 𝑆

Example

Let $𝑆$ be the set of real numbers whose cosine is greater than $\frac{1}{2}$ :

𝑆 = {𝑥 \in ℝ : cos (𝑥) > \frac{1}{2}}

Universal Sets and Complements

$Ω$ : universal set (sample space)

Compliment of a set

𝑥 \in 𝑆^{𝑐} iff 𝑥 \in Ω and 𝑥 \notin 𝑆

Key identities:

{(𝑆^{𝑐})}^{𝑐} = 𝑆

$\emptyset$ : the empty set

Ω^{𝑐} = \emptyset

Subsets

If set $𝑆$ is a subset of the set $𝑇$ :

𝑆 \subset 𝑇

This means:

𝑥 \in 𝑆 \Rightarrow 𝑥 \in 𝑇

Union

𝑆 \cup 𝑇

Membership Rule

𝑥 \in 𝑆 \cup 𝑇 \Leftrightarrow 𝑥 \in 𝑆 \lor 𝑥 \in 𝑇

Intersection

𝑆 \cap 𝑇

Membership Rule

𝑥 \in 𝑆 \cap 𝑇 \Leftrightarrow 𝑥 \in 𝑆 \land 𝑥 \in 𝑇

Let

𝑆_{𝑛} 𝑛 = 1, 2, \dots

Countable Union

𝑥 \in ⋃_{𝑛} 𝑆_{𝑛} iff 𝑥 \in 𝑆_{𝑛} 𝐟𝐨𝐫 𝐬𝐨𝐦𝐞 𝑛

Countable Intersection

𝑥 \in ⋂_{𝑛} 𝑆_{𝑛} iff 𝑥 \in 𝑆_{𝑛} 𝐟𝐨𝐫 𝐚𝐥𝐥 𝑛

Set properties