Sets

Definition

A set is a collection of well defined distinct objects.
These objects are called members or elements of the set.

Ways to Describe a Set

The ways to describe a set are following

Descriptive Method

A set may be described in terms of English words.

Example

Set of all cities in Pakistan.

Tabular Method

A set may be described by listing all of its elements.

Example

\[\mathbb{N} = \{1, 2, 3, \ldots \}\]

Set Builder Method

A set maybe be described by stating the symbols and describing the property that is common between all elements.

Example

\[\mathbb{W} = \{x | x \ge 0 \land x < \infty\}\]

Subsets and Supersets

Sets can contain other sets.
Let us take an example of \(\mathbb{N}\) and \(\mathbb{Z}\) sets.

\[\forall x \in \mathbb{N} \implies x \in \mathbb{Z} \implies \mathbb{N} \subset \mathbb{Z}\]

Here we say that \(\mathbb{N}\) is a subset of \(\mathbb{Z}\).
We can also say that \(\mathbb{Z}\) is a superset of \(\mathbb{N}\)

\[\mathbb{Z} \supset \mathbb{N}\]

Operations on Set

Let's understand with a example.
Let's take 2 sets, A and B for example.

\[A = \{1, 2\}\]

\[B = \{2, 3\}\]

Union

Union is a binary operator which takes 2 sets as input and then produces another set as output which contains elements which belong to either set.

\[A \cup B = \{1, 2, 3\}\]

Intersection

Intersection is also a binary operator which takes 2 sets as input and then produces another set as output which contain only those elements which are common to both sets.

\[A \cap B = \{2\}\]