Venn diagrams

Venn diagrams introduced by John venn represents relationship between finite collection of sets.
Consider two sets A,B.
In venn diagrams they are represented as

Logic and Venn diagrams
AUB

A∩B

(AUB)'

(A∩B)'

A'UB

A'∩B

Example
A group of 40 students were surveyed about the languages they have chosen E = English, H = Hindi, T = Tamil.
3 students did not study any of the languages above.
2 students study all three languages
8 study English and Hindi
10 study English and Tamil
6 study Hindi and Tamil
13 students study Hindi
28 students study English
(a) Draw a Venn diagram to illustrate the data given above
(b) How many students study only Tamil?
(c) On your diagram represent the students who do not study English or Hindi.

Solution
A
• There are 3 sets within a universal set.Draw a Venn diagram with 3 intersecting sets.
• The 3 students who do not study any language go inside the box, but outside of the circles.
• 2 students must go in the intersection of all three circles.
• 8 must go in the intersection of E and H, but remember that 2 have already been filled in the intersection of all 3 sets. So only 6 will be put in the intersection.
• Likewise the English and Tamil intersection will have 10 – 2 = 8 and only 4 will go in the Hindi and Tamil intersection.
• In the Hindi set there should be 13, but thus far we have filled in 6+2+4 = 12. So only 1 student is left to go in the Hindi only set.
• Likewise 28 students study English, but we have filled in 16 of these students, so 12 go in the English only section.
• There are 40 students altogether . Counting all the numbers filled in thus far reveals we have put 36 numbers on the diagram. It is important to remember the 3 outside the circles, but in the box. Fill in 4 in the Tamil only section.
• Finally the diagram will be like this,

B
Students reading only Tamil is 4
C