Discrete Mathematics Answers

E. What rule of inference is used in each of the following arguments? Show solution.

1. If it will rain today, then the classes are suspended. The classes are not suspended today. Therefore, it did not rain today.

2. If you read your module today, then you will not play ML today. If you cannot play ML today, you can play ML tomorrow. Therefore, you read

your module today, then you will play ML tomorrow. 

PROPOSITIONAL LOGIC:

Let p, q, and r denote the following statements:

p: A square has four equal side

q: Rectangle has 2 parallel sides

r: A square is a rectangle.

1. Express each of the following into an English sentence. (3 pts each)

a. r ^ q → p

b. p̅ → q

c. q → p̅ v r

2. Write T if the above item is true and F if it is false. Show solution. (3 pts

each)

a.

b.

c. 

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

As you now that

0^{3 0}

1^{3 1}

2^{3 8}

3^{3 27}

4^{3 64}

5^{3 125}

so, we have a proof that every positive integer is the sum of the cubes of eight non negative integers. For example

7= 1³ + 1³ + 1³ +1³+1³+1³+ 1³ +0³

12= 2³+ 1³ + 1³ + 1³ +1³+0³+0³+0³

Can you disprove this statement?

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

Proof by contradiction that if n is a positive integer, then n is odd if and only 

if 5n + 6 is odd

 Suppose you have a graph with v vertices and e edges that satisfies v=e+1. Must the graph be a tree? Prove your answer.