Discrete Mathematics Answers

show that

1²+3²⁺5²+.....(2n+1)²=(n+1)(2n+1)(2n+3)/3

where n is a nonnegative integer.

Let R1 and R2 be symmetric relations. Is R1 ∩ R2 also symmetric? Is R1 ∪ R2 also

symmetric?

Determine whether each of these functions is a bijection from R to R (real).a) f (x) = 2x + 1b) f (x) = x2 + 1c) f (x) = x3

Draw the Hasse diagrams of all partial ordered sets with at most 4 elements. Which of these are lattices?

Given the following 2 premises, 1. 𝑝→(𝑞∨𝑟)

2. 𝑞→𝑠

Prove 𝑝→(𝑟∨𝑠) is valid using the Proof by Contradiction method.

An algorithm is a _________ set of precise instructions for performing computation.

show that factorial function is promitive recursive

4. Let P(n) be the statement that 1+2+...+n'3D (n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis. f) Explain why these steps show that this formula is true whenever n is a positive integer. 5. Prove that 12+32+52+ + (2n + 1)² = (n + 1)

4. Let P(n) be the statement that 1+2+...+n'3D (n(n + 1)/2)2 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the proof. c) What is the inductive hypothesis? d) What do you need to prove in the inductive step? e) Complete the inductive step, identifying where you use the inductive hypothesis. f) Explain why these steps show that this formula is true whenever n is a positive integer. 5. Prove that 12+32+52+ + (2n + 1)² = (n + 1)

1.     Write the multisets (bags) of prime factors of given numbers.

i.       160

ii.      120

iii.    250

2.     Write the multiplicities of each element of multisets (bags) in Part 2-1(i,ii,iii) separately.

3.     Determine the cardinalities of each multiset (bag) in Part 2-1(i,ii,iii).