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20 people applied for a job. Everyone either has a school certificate or diploma or even both. If 14 have school certificates and 11 diplomas how many have a school certificate only

Sagot :

Given:

Either has a school certificate or diploma or even both = 20 people

Having school certificates = 14

Having diplomas = 11

To find:

The number of people who have a school certificate only.

Solution:

Let A be the set of people who have school certificates and B be the set of people who have diplomas.

According to the given information, we have

[tex]n(A)=14[/tex]

[tex]n(B)=11[/tex]

[tex]n(A\cup B)=20[/tex]

We know that,

[tex]n(A\cup B)=n(A)+n(B)-n(A\cap B)[/tex]

[tex]20=14+11-n(A\cap B)[/tex]

[tex]20=25-n(A\cap B)[/tex]

Subtract both sides by 25.

[tex]20-25=-n(A\cap B)[/tex]

[tex]-5=-n(A\cap B)[/tex]

[tex]5=n(A\cap B)[/tex]

We need to find the number of people who have a school certificate only, i.e. [tex]n(A\cap B')[/tex].

[tex]n(A\cap B')=n(A)-n(A\cap B)[/tex]

[tex]n(A\cap B')=14-5[/tex]

[tex]n(A\cap B')=9[/tex]

Therefore, 9 people have a school certificate only.