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\;-\;AExample:
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\;=\;UThe 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.
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