### Theory:

Set operation is an operation used to construct new sets from given sets.
There are several fundamental operations for constructing new sets from given sets. They are:
• Universal set.
• Union of two sets.
• Intersection of two sets.
• Difference of two sets.
• Symmetric difference of sets.
• Complement sets.
• Disjoint sets.
• Overlapping sets.
Universal set:
A universal set is the set of all elements under consideration, denoted by capital $$U$$. All other sets are subsets of the universal set.

In the Venn diagram below, represents a universal set.

Union of two sets:
Let $$A$$ and $$B$$ be any two sets. The union of $$A$$ and $$B$$ is the set consists of all elements of $$A$$ and $$B$$; common elements being taken only once. The symbol $$∪$$ is used to denote the union. Symbolically, we write $$A ∪ B$$ and usually read as $$A$$ union $$B$$.

Symbolically, we write $$A\cup B = \{x: x\in A\ \mathrm{or}\ x\in B\}$$

Let $$A=\{2,\ 4,\ 6,\ 8\}$$ and $$B=\{6,\ 8,\ 10,\ 12\}$$.

Then, $$A\cup B=\{2,\ 4,\ 6,\ 8,\ 10,\ 12\}$$

Note: Common elements $$6$$ and $$8$$ have been taken only once while writing $$A ∪ B$$.

In the Venn diagram below $$A$$ and $$B$$ are represented.

In the Venn diagram below, the area in the blue colour represents $$A ∪ B$$.

Intersection of two sets:
The intersection of sets $$A$$ and $$B$$ is the set of elements which are common in $$A$$ and $$B$$. The symbol $$∩$$ is used to denote intersection.

Symbolically, we write $$A\cap B=\{x:x\in A\ \mathrm{and}\ x\in B\}$$.

Numerical: Let $$A=\{2,\ 4,\ 6,\ 8\}$$ and $$B=\{6,\ 8,\ 10,\ 12\}$$.

Then, $$A\cap B=\{6,\ 8\}$$.

In the Venn diagram below $$A$$ and $$B$$ are represented.

In the Venn diagram below, the area in the blue colour represents $$A ∩ B$$.

Important!
The relation "is an element of", also called set membership, is denoted by the symbol "$$∈$$", and the relation "is not an element of", is denoted by the symbol "$$∉$$".