Question

1. Solve the following two " union " type questions:
(a) How many bit strings of length 9 either begin with 2 0s or end with 2 1s? (inclusive or)

(b) Every student in a discrete math class is either a computer science or a mathematics major or is a joint major in these two subjects. How many students are in the class if there are 30

computer science majors (including joint majors), 20 math majors (including joint majors) and 10 joint majors?

2. How many elements are in the union of four sets if each of the sets has 99 elements, each pair of sets share 51 elements, each triple of sets shares 24 elements and there are 2 elements in all four sets.
3. One of Shakespeare's sonnets has a verb in 12 of its 15 lines, an adjective in 9 lines, and both in 7 lines.
How many lines have a verb but no adjective?
How many lines have an adjective but no verb?
How many have neither an adjective nor a verb?

Answer 1

Answer:

2.

Consider,

P(A U B U C U D) = P(A) + P(B) + P(C) + P(D) - P(A ⋂ B) - P(B ⋂ C) - P(C ⋂ D) - P(D ⋂ A) + P(B ⋂ C ⋂ D) + P(C ⋂ D ⋂ A) + P(C ⋂ D⋂ A) + P(D ⋂ A ⋂ B) - P(A ⋂ B ⋂ C ⋂ D) - P(A ⋂ C) - P(B ⋂ D)

substitute the given values

= 4*99 - 6*51 + 4*24 - 2

= 396 - 306 + 96 - 2

= 184

So there are 184 elements in the union set.

3.

Given,

n(u) = 15

n(A) = 12

n(B) = 9

n(A ⋂ B) = 7

a)

To give number of lines have a verb but no adjective n(A - B)

i.e.,

= n(A) - n(A ⋂ B)

substitute values

= 12 - 7

= 5

b)

To give number of lines have an adjective but no verb

n(B - A) = n(B) - n(A ⋂ B)

= 9 - 7

= 2

c)

To give neither an adjective nor a verb

i.e.,

n(u) - n(A U B)

= 15 - (12 + 9 - 7)

= 15 - 14

= 1

I hope it works for you,

Please post the 1st question as separate post. Thank you.