Discrete Mathematics Answers

Generate up to the seventh and nineth rows of the Pascal triangle.

Prove by induction that P(n): 2+3+3...+n=n(n+1)/2 Æn ≥ 1

Give a contrapositive proof of the theorem; "If n is an interfer and 3n + 2 is even, then n is even."?

verify each of the following equivalences using basic equivalences

1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S

2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Draw the Hasse diagram for divisibility on the set, {1, 2, 3, 4, 5, 6, 7, 8}?

RULE OF INFERENCE

A. What rule of inference is used in each of the following arguments?

Show solution. (5 pts each)

1. If I will read my modules, then I can answer all the activities. If I can answer all the activities, then I will get high scores. Therefore, if I will read my modules, then I will get high scores.

2. Rizza is an IT student. Therefore, Rizza is either an IT student or a

programmer.

3. If it is a national holiday, then school is closed. It is a national holiday.

Therefore, the school is closed.

4. If Ann does not love numbers or if Ann does not love programming.

If Ann loves numbers, then she can be a mathematician. Therefore,

Ann can be a mathematician.

Represent the following relation to a directed graph. relation R = {(1, 1), (1,2), (3, 2)} on set S = {1,2,3},}

Use the truth table to transform each of the following wffs into the full conjunctive normal form

(P→Q)→P. P→(Q→P)

(P∨Q)∧R. P→Q∧R. Q∧¬P→P

verify each of the following equivalences using basic equivalences

1)((P∧Q∧R)→S∧(R→(P ∨ Q ∨ S))≡R∧(P↔Q)→S

2)((P∧Q)→R)∧(Q→(S∨R))≡Q∧(S→P)→R

Show that if x is an integer then x2+x-41= 0 produce prime numbers