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.
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-16-51-38.png)
Venn\;Diagram\;A\;\subseteq\;U
U is called the universal set and it contains everything.
A\;\subseteq\;U
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-16-56-56.png)
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
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-19-52-35-1024x693.png)
The area in red represents the complement of A, Ā.
Venn diagram of A ∪ B
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-19-56-14-1024x622.png)
The red area represents the union of A and B.
Venn diagram of A ∩ B
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-19-57-33-1024x637.png)
The red area represents the intersection of A and B.
Venn Diagram for A – B
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-19-58-29-1024x613.png)
The red area represents the set difference between A and B.
Venn Diagram A ⊕ B
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-19-59-40.png)
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.
![](https://roselearnstocode.com/wp-content/uploads/2023/10/Screenshot-at-Oct-24-20-05-24-1024x751.png)
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