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In a class of 80 students,40 study physics,48study mathematics and 44 study chemistry,20 study physics and mathematics,24 study physics and chemistry and 32 study only two of the three subjects.Of every student studies at least one of the three subjects.Find ; 1.The number of students who study all the three subjects.2.The number of students who study only mathematics and chemistry.

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Answer:

80+40+44-20Ă·24+32 ♡♡♡

Using set operations,

1.The number of students who study all the three subjects = 10

2.The number of students who study only mathematics and chemistry = 8

What are set operations?

Set operations are the operations that are applied on two or more sets to develop a relationship between them. There are four main kinds of set operations which are as follows.

  • Union of sets
  • Intersection of sets
  • Complement of a set
  • Relative Complement

Let mathematics = M, physics = P, chemistry = C.

n(MâˆȘPâˆȘC) = 80

n(M) = 48

n(P) = 40

n(C) = 44

n(M∩P) = 20

n(M∩C) = ?

n(C∩P) = 24

n(M∩P) + n(M∩C) + n(C∩P) - 3n(C∩P∩M) = 32

20 + n(M∩C) + 24 - 3n(C∩P∩M) = 32

n(M∩C) = 3n(C∩P∩M) - 12 - *

We know,

n(MâˆȘPâˆȘC) = n(M) + n(P) + n(C) - n(M∩P) - n(M∩C) - n(C∩P) + n(C∩P∩M)

80 = 48 + 40 + 44 - 20 - n(M∩C) - 24 + n(C∩P∩M)

n(M∩C) = n(C∩P∩M) + 8 - **

Equating * and **,

n(C∩P∩M) + 8 = 3n(C∩P∩M) - 12

2n(C∩P∩M) = 20

n(C∩P∩M) = 10

putting the value of n(C∩P∩M) in *

n(M∩C) = 18

n(M∩C) - n(C∩P∩M) = 8

Learn more about set operations here

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