Study Card - Set Theory and Venn Diagrams

Please use your left and right arrow keys to navigate.

Use only latest version of browsers like Firefox or Chrome or Opera

Introduction

A Set is a group of things.

Venn diagrams represent relationships between two or more sets.

Intersection of Sets

Union of Sets

Cardinality

Formulas to remember

Formula 1:

Formula 2

Example 1:

In a village, 70% of the people like tea, 65% like coffee, 27% do not like tea or coffee. If 248 people drink both tea and coffee, find the total number of people in the village.

Let us draw the Venn diagram.

Solution:

% people who drink tea or coffee = n(tea U coffee) = 100% - 27% = 73%

n(tea U coffee) = n(tea) + n(coffee) - n (tea n coffee)

73% = 70% + 65% - n (tea n coffee)

n (tea n coffee) = 62%

Let the total number of people in the village be y.

0.62y = 248.

Hence y=248/0.62 = 248*100/62 = 400.

Total number of people in the village = 400.

Example 2

In a group of 20 students there are 8 females, 9 engineers and 6 female engineers. Find the number of male non-engineers in the group.

Let us draw the Venn diagram.

Solution:

Number of males = 20-8 = 12

Number of male engineers = Total number of engineers – Female engineers = 9-6 =3

Hence number of male non engineers = Number of males - Number of male engineers

= 12 – 3 = 9

Example 3

In a town 65% people watched the news on television , 40% read a newspaper and 25% read a newspaper as well as watched the news on television. What percent of the people neither watched the news on television nor read a news paper?

Let us draw the Venn diagram.

Solution:

People who read news paper or watch TV = n (TV) + n(News Paper) – n (who watch TV and reads news paper)

= 65% + 40 % - 25% = 80%

Hence percent of the people who neither watch the news on television nor read news paper = 100% - 80% = 20%.

Example 4

Among one hundred applicants for a certain teaching position, ten had never taken a course in chemistry or in physics. Seventy-five had taken at least one course in chemistry. Eighty-three had taken at least one course in physics. How many of the applicants had experience in both chemistry and physics?

Let us draw the Venn diagram.

Solution:

Ten had never taken a course in chemistry or in physics.

Hence 100-10 = 90 had taken a course either in chemistry or in physics.

n(course either in chemistry OR in physics) = n(course in chemistry) + n(course in physics) – n(course in both chemistry AND physics)

90 = 75 + 83 – n(course in both chemistry AND physics)

n(course in both chemistry AND physics) = 158 – 90 =68

Applicants who had experience in both chemistry and physics=68

Example 5

A survey on a sample of 25 new phones was conducted to see which of the three options wifi, video recording and voice recording were present. The survey found:

• 15 had wifi.

• 2 had wifi and voice recording but no video recording.

• 12 had video recording.

• 6 had wifi and video recording but no voice recording.

• 11 had voice recording.

• 1 had video recording and voice recording but no wifi.

• 3 had all three options.

What is the number of phones that had none of the three options?

Let us draw the Venn diagram.

Solution:

Continued on next page ......

Solution for Example 5