Venn Diagrams: CAT Quantitative Aptitude

A set can be represented by closed figures like circles, triangles, rectangles etc. The point in the interior of the figure represents the elements of the set. Such a representation is called a Venn diagram.

Examples:

(i)   B   A can be represented as

(ii)  A° can be represented as.

(iii) A =  All students in Mumbai
      B  = Students in one of the schools in Mumbai
      C  = Students in Grade XII of that school

Similarly, one can work out the Venn Diagram notation for Union of Sets, Intersection of Sets, Disjoint Sets, Distributive Property of Union and Intersection of Sets, Difference of Sets etc.

The best way to understand the application is to go through some solved examples as shown below.

Example:

The market research of a certain breakfast cereal firm interviewing 100 people, found that on a certain morning for breakfast 72 had cereals, 39 had an egg, 75 had toast; 32 had cereal and egg, 53 had cereal and toast, 26 had toast and egg. Of these 21 had cereal, toast and egg. How many of those interviewed had neither cereal, toast nor egg?

[1] 6
[2] 4
[3] 5
[4] 8

Solution:

Let, the number of people who had cereal be denoted by n(C)
Similarly, let n(E), n(T), n(C U E U T) denote the number of people who had egg, toast and all three i.e., cereal, egg and toast, respectively.

Also, let n(C ∩ T), n(E ∩ T), n(C ∩ E), n(C ∩ T ∩ E) denote the number of people who — neither had cereal nor toast; neither had egg nor toast; neither had cereal nor egg; neither had cereal, toast nor egg, respectively.

Then,

n(C U E U T) = n(C) + n(E) + n(T) – n(C ∩ T) – n(E ∩ T) – n(C ∩ E) + n(C ∩ T ∩ E)

                  =  72    + 39    + 75   – 53   – 26   – 32   + 21
                  = 96

n(C U E U T)’ = 100 – 96 = 4

4 people had neither cereal, toast nor egg. Hence, [2] is the correct option.

Now, try to show the above using Venn Diagrams.