# Venn Diagrams in Different Situations

To draw Venn diagrams in different situations are discussed below:

How to represent a set using Venn diagrams in different situations?

1. ξ is a universal set and A is a subset of the universal set.

ξ = {1, 2, 3, 4}

A = {2, 3}

Draw a rectangle which represents the universal set.

Draw a circle inside the rectangle which represents A.

Write the elements of A inside the circle.

Write the leftover elements in ξ that is outside the circle but inside the rectangle.

Shaded portion represents A’, i.e., A’ = {1, 4}

2. ξ is a universal set. A and B are two disjoint sets but the subset of the universal set i.e., A ⊆ ξ, B ⊆ ξ and A ∩ B = ф

For example;

ξ = {a, e, i, o, u}

A = {a, i}

B = {e, u}

Draw a rectangle which represents the universal set.

Draw two circles inside the rectangle which represents A and B.

The circles do not overlap.

Write the elements of A inside the circle A and the elements of B inside the circle B of ξ.

Write the leftover elements in ξ , i.e., outside both circles but inside the rectangle.

The figure represents A ∩ B = ф

3. ξ is a universal set. A and B are subsets of ξ. They are also overlapping sets.

For example;

Let ξ = {1, 2, 3, 4, 5, 6, 7}

A = {2, 4, 6, 5} and B = {1, 2, 3, 5}

Then A ∩ B = {2, 5}

Draw a rectangle which represents a universal set.

Draw two circles inside the rectangle which represents A and B.

The circles overlap.

Write the elements of A and B in the respective circles such that common elements are written in overlapping portion (2, 5).

Write rest of the elements in the rectangle but outside the two circles.

The figure represents A ∩ B = {2, 5}

4. ξ is a universal set and A and B are two sets such that A is a subset of B and B is a subset of ξ.

For example;

Let ξ = {1, 3, 5, 7, 9}

A= {3, 5} and B= {1, 3, 5}

Then A ⊆ B and B ⊆ ξ

Draw a rectangle which represents the universal set.

Draw two circles such that circle A is inside circle B as A ⊆ B.

Write the elements of A in the innermost circle.

Write the remaining elements of B outside the circle A but inside the circle B.

The leftover elements of are written inside the rectangle but outside the two circles.

Observe the Venn diagrams. The shaded portion represents the following sets.

(a) A’ (A dash)

(b) A ∪ B (A union B)

(c) A ∩ B (A intersection B)

(d) (A ∪ B)’ (A union B dash)

(e) (A ∩ B)’ (A intersection B dash)

(f) B’ (B dash)

(g) A - B (A minus B)

(h) (A - B)’ (Dash of sets A minus B)

(i) (A ⊂ B)’ (Dash of A subset B)

For example;

Use Venn diagrams in different situations to find the following sets.

(a) A ∪ B

(b) A ∩ B

(c) A'

(d) B - A

(e) (A ∩ B)'

(f) (A ∪ B)'

Solution:

ξ = {a, b, c, d, e, f, g, h, i, j}

A = {a, b, c, d, f}

B = {d, f, e, g}

A ∪ B = {elements which are in A or in B or in both}

= {a, b, c, d, e, f, g}

A ∩ B = {elements which are common to both A and B}

= {d, f}

A' = {elements of ξ, which are not in A}

= {e, g, h, i, j}

B - A = {elements which are in B but not in A}

= {e, g}

(A ∩ B)' = {elements of ξ which are not in A ∩ B}

= {a, b, c, e, g, h, i, j}

(A ∪ B)' = {elements of ξ which are not in A ∪ B}

= {h, i, j}

Set Theory