# 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.**

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.**

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)

(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}