please help me with the solution

Question

b) Each student in liberal arts at some college has a mathematics requirement A and a science requirement B. a poll of 140 students shows that; 60 completed A, 45 completed B, 20 completed both A and B. use a venn diagram to find the number of students who have completed
i. At least one of A and B
ii. Exactly one of and B
iii. Neither A nor B

Answer 1

If 60 students have completed the math requirement, 45 students have completed the science requirement and 20 have completed both, then.

 

40 have completed only math

25 have completed only science

20 have completed both

85 subtotal

 

Note I subtracted 20, the number of students who have already completed both requirements from both the number of students who have completed math and the number of students who have completed science.

 

Of the 140 students polled, this leaves 55 students (140-85) who have not yet completed either requirement.

 

As to the questions

i. # students who have completed at least one of A and B.

This means how many students have either completed the math requirement, the science requirement or both.

Ans:  85 students

 

ii. # students who have complete exactly one of A and B

Here we want the number of students who completed math requirement OR science requirement, but not both.

Ans: 40+25=65 students

 

iii. # students who have not completed either A or B

Ans: 140-85 = 55 students