# Day 53 – Set Representation and Manipulation

## Universal Set

A universal set is a set that contains everything. We note the universal set with the letter U.

## Venn Diagrams

A Venn diagram is used to visualize the possible relations among a collection of sets.

Venn\;Diagram\;A\;\subseteq\;U

U is called the universal set and it contains everything.

A\;\subseteq\;U

Supporting:

## Complement of a set

The complement of a set A, Ā, contains all the elements in the universal set U but not in A.

\overline{A}\;=\;U\;-\;A

Example:

U\;=\;\{1,2,3,4,5,6,7,8,9,10\}\;\;and\;\;A\;=\;\{2,4,6,8,10\}\\then\\\overline{A}\;=\;\{1,3,5,7,9\}
\overline{A}\;\cup\;A\;=\;U

The union of a set A and its corresponding complement is always equal to the universal set.

## Venn diagram for Ā

The area in red represents the complement of A, Ā.

## Venn diagram of A ∪ B

The red area represents the union of A and B.

## Venn diagram of A ∩ B

The red area represents the intersection of A and B.

## Venn Diagram for A – B

The red area represents the set difference between A and B.

## Venn Diagram A ⊕ B

The red area represents the symmetric difference between A and B.

## Example:

A\;\oplus\;B\;=\;A\;\cup\;B\;-\;(A\;\cap\;B)

The symmetric difference between A and B is equals to the union of A and B less than the intersection of A and B.

## De Morgan’s Law

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