Discrete Mathematics Answers

Consider the following sets.

U = { 1, 2, …. 10 }

A = { 3, 4, 5, 6, 8, 10 } ` C = { 2, 4, 6, 7, 8 }

B = { 1, 3, 5 }

Perform the following set operations by writing the resulting sets using the roster method. Moreover, sketch the Venn diagram.

1. A’ = ________________________________________________

2.B’ = ________________________________________________

3. A U B = _____________________________________________

4. A ∩ B = _____________________________________________

5. A' ∩ B = _____________________________________________

6. (A ∩ B) U C = _________________________________________

7. A ∩ (B U C’) = _________________________________________

Using a Venn diagram, show the following:

1. A U B

2. (A U B)’

3. A ∩ B’

4. A’ U B’

5. A ∩ B ∩ C

Simplify (A ∩ B) U (A ∩ B’) U (A’ ∩ B) U (A’ ∩ B’)

Qn 3. Consider a sequence {an}∞

n=0 of real numbers defined as follows:

a0 = 3, a1 = 2, a2 = 12, and

2an+3 − an+2 − 8an+1 + 4an = 0 for all n ≥ 0.

(i) What is a99?

(ii) Find a formula for 2a2n+1 − a2n in terms of n

Mark is shopping for a new mobile phone. The

probability that someone buys Oppo is 25%,

Huawei is 20%, Vivo is 10%, both Oppo and

Vivo is 5%, both Oppo and Huawei is 7%,

both Vivo and Huawei is 3%, and of the three

brands is 2%. What is the probability that

Mark buys Oppo or Huawei?

Of 2000 persons under investigation (PUI) due

to Covid-19, 1050 PUI undergone rapid

testing, 500 PUI undergone swabtesting, and

200 PUI undergone both rapid and swab

testing. What is the probability that a

randomly selected PUI undergone random

testing or swab testing?

10. Find the coefficient of x²⁰ in A(x)=(x¹⁰+x¹¹+.....+x²⁵)(x¹+x²+....+x¹⁵)⁵(x²⁰+x²¹+.....+x⁴⁵)

9. In the expression (1+x⁵+x⁹)¹⁰ find the coefficient of a) x²³ b) x³²

6. Team A has probability 2/3 of winning whenever it plays.suppose A plays four Games.find the probability p that A wins more than half of its gamed.

Prove that if 30 dictionaries in a library contains a total of 61,327 pages, then at least one of

the dictionaries must have at least 2045 pages.