Discrete Mathematics Answers

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).

How to draw a network diagram in mathematical foundations of computer science (10mark)

Use the Principle of Mathematical Induction to prove that "((2n) !)\/(2^n n !)" is odd for all positive integers.

Justify whether the given operations on relevant sets are binary operations or not.

- Multiplication and Division on set of natural numbers

- Subtraction and Addition on Set of natural numbers

- Exponential operation: (x, y) → xy on Set of Natural numbers and set of Integers.

(p^q)v(P ^ Q ^ R)

Suppose that A,B and C are sets such that A is the improper subset of B and B is the improper subset of C