3. Venn Diagram and Set Operations

 

Venn Diagrams

Venn diagram is a pictorial representation of sets in which sets are represented by enclosed areas in the plane.

The universal set U is represented by the interior of a rectangle, and the other sets are represented by disks lying within the rectangle.

Partitions of a Set

Let S be a non-empty set. A partition of S is a subdivision of S into non-overlapping, non-empty subsets. Specifically, a partition of S is a collection {A_n} of non-empty subsets of S such that :

• Each element in S belongs to one of the A_n.
• The sets of {A_n} are mutually disjoint, i.e. A_i ≠ A_j, then A_i∩A_j = ∅.

The subsets in a partitions are called cells.

Operations on Sets

Some of the basic operations on sets are:

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.

A∪B = {x : x ∈ A or x ∈ B}

Example: Let A = {a, b, c, d, e}, B = {a, e, i, o, u}
then A∪B = {a, b, c, d, e, i, o, u}

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.

A∩B = {x : x ∈ A and x ∈ B}

Example: Let A = {a, b, c, d, e}, B = {a, e, i, o, u}
then A∩B = {a, e}

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 A and is denoted by A^', A^c or Ā

A^c = U-A = {x : x ∈ U and x ∉ A}

Example: Let U = {1, 2 , 3, 4, 5, 6} and A = {1, 3, 5}
then A^' or A^c = {2, 4, 6}

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 or A∖B.

A-B = {x : x ∈ A and x ∉ B}

Example: Let A = {11, 12, 13, 14, 15} and B = {10, 12, 14, 16, 18}
then A-B = {11, 13, 15}, B-A = {10, 16, 18}

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 or AΔB.

A⊕B = {x : (x ∈ A and x ∉ B) or (x ∉ A and x ∈ B)}
A⊕B = (A-B) ∪ (B-A)
A⊕B = (A∪B) - (A∩B)

Example: Let A = {2, 4 ,5, 8} and B = {2, 6, 7 ,8}
then A⊕B or AΔB = {4, 5, 6, 7}

Previous                                 Next