Discrete Mathematics Answers

Show that each of these conditional statements is a tautology by using truth tables.

a) (p ∧ q) → p                b) p → (p ∨ q)

c) ￢p → (p → q)                            d) (p ∧ q) → (p → q)

e) ￢(p → q) → p                        f) ￢(p → q) → ￢q

Show that (p → r) ∧ (q → r) and (p ∨ q) → r are logically equivalent.

Consider the following series 56, 28, 14..

I. Find 17th term.

ii. Find the sum of the series if it continues indefinitely

iii. Find 20th term.

A house which is valued at 2,600, 000 appreciated at rate of 12% per year

I. What will be its value after two and half years?

ii. After how long will its value be 4200,000

Write each of these statements in the form “if p, then q” in English.

a) It snows whenever the wind blows from the northeast.

b) The apple trees will bloom if it stays warm for a week.

c) That the Pistons win the championship implies that they beat the Lakers.

d) It is necessary to walk eight miles to get to the top of Long’s Peak.

e) To get tenure as a professor, it is sufficient to be world famous.

f ) If you drive more than 400 miles, you will need to buy gasoline.

g) Your guarantee is good only if you bought your CD player less than 90 days ago.

h) Jan will go swimming unless the water is too cold.

i) We will have a future, provided that people believe in science.

Write each of these statements in the form “if p, then q” in English.

a) It snows whenever the wind blows from the northeast.

b) The apple trees will bloom if it stays warm for a week.

c) That the Pistons win the championship implies that they beat the Lakers.

d) It is necessary to walk eight miles to get to the top of Long’s Peak.

e) To get tenure as a professor, it is sufficient to be world famous.

f ) If you drive more than 400 miles, you will need to buy gasoline.

g) Your guarantee is good only if you bought your CD player less than 90 days ago.

h) Jan will go swimming unless the water is too cold.

i) We will have a future, provided that people believe in science.

Use generating functions to solve the recurrence relation an = 4a_n−1 − 4a_n−2 +n²

, where a₀ = 2, a₁ = 5.

Define Semigroup and Monoid. Show that the set of positive Integer is a monoid for the operation

defined by aOb = max{ a,b}

Let the operations * and ⊕ be defined on the set of integers.

Find each of the following are commutives, associatives, left-distributive, and right-distributive.

1.   a*b=ab; a⊕b=-a-b

2.   a*b=a+2b; a⊕b=a-2b

3.   a*b=2ab; a⊕b=2a+2b+4ab

4.   a*b=-a-b-2ab; a⊕b=3a+3b

5.   x*y=x²+2x+y²; x⊕y=x+y

Show that a tree with n vertices has exactly n-1 edges