Operations on Sets

The basic set operations are:

1. Union of Sets: Union of Sets A and B is defined to be the set of all those elements which belong to A or B or both and is denoted by A∪B.

1. A∪B = {x: x ∈ A or x ∈ B}  

Example: Let A = {1, 2, 3},       B= {3, 4, 5, 6}
A∪B = {1, 2, 3, 4, 5, 6}.

2. Intersection of Sets: Intersection of two sets A and B is the set of all those elements which belong to both A and B and is denoted by A ∩ B.

1. A ∩ B = {x: x ∈ A and x ∈ B}  

Example: Let A = {11, 12, 13},       B = {13, 14, 15}
A ∩ B = {13}.

3. Difference of Sets: The difference of two sets A and B is a set of all those elements which belongs to A but do not belong to B and is denoted by A - B.

1. A - B = {x: x ∈ A and x ∉ B}  

Example: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A - B = {3, 4} and B - A = {5, 6}

4. Complement of a Set: The Complement of a Set A is a set of all those elements of the universal set which do not belong to A and is denoted by A^c.

Ac = U - A = {x: x ∈ U and x ∉ A} = {x: x ∉ A}

Example: Let U is the set of all natural numbers.
A = {1, 2, 3}
A^c = {all natural numbers except 1, 2, and 3}.

5. Symmetric Difference of Sets: The symmetric difference of two sets A and B is the set containing all the elements that are in A or B but not in both and is denoted by A ⨁ B i.e.

1. A ⨁ B = (A ∪ B) - (A ∩ B)  

Example: Let A = {a, b, c, d}
B = {a, b, l, m}
A ⨁ B = {c, d, l, m}