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.

$Subset of the Universal Set$

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

$Two Disjoint Sets$

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.

$Overlapping Sets$

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.

$A is a Subset of B$

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)

$A dash Set$

(b) A ∪ B (A union B)

$A union B$

$A intersection B$

(d) (A ∪ B)’ (A union B dash)

$A union B dash$

(e) (A ∩ B)’ (A intersection B dash)

$A intersection B dash$

(f) B’ (B dash)

$B dash$

(g) A - B (A minus B)

$A minus B$

(h) (A - B)’ (Dash of sets A minus B)

$Dash of sets A minus B$

(i) (A ⊂ B)’ (Dash of A subset B)

$Dash of A subset B$

For example;

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

$Venn Diagrams in Different Situations$

(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}