Discrete Mathematics Answers

Prove that the complete disjunctive normal form and complete conjunctive normal form in three variables in a Boolean algebra is equal to the unit and zero element of the Boolean algebra.

List the members of the set

(x l x is an integer such that x < 2)

In a party, there are 12 people with white shirts and 7 people with red shirts;

4 people have black shoes and white shirts;

5 people have black shoes and red shirts;

the total number of people with white or red shirts or black shoes is 21.

How many people have black shoes?

In a party, there are 12 people with white shirts and 7 people with red shirts;

4 people have black shoes and white shirts;

5 people have black shoes and red shirts;

the total number of people with white or red shirts or black shoes is 21.

How many people have black shoes?

there are 450 farmers. 250 pounds of farm beetroot, 110 pounds of farm yams, 75 pounds of farm radish, 45 pounds of farm beetroot and radish, 40 pounds of farm yams and radish, and 30 pounds of farm beetroot and yams Let B, Y, and R represent the farms that grow beets, yams, and radish, respectively.

Determine the number of farmers that farm beetroot, yams, and radish.

Consider a graph where V(G)={1, 2, 3, 4} and E(G)=[{1,2}, (1,2), {1,4}, {2,3}, {3,4}, {3,4}]. How many Hamilton cycles does it have?

Consider the following relations on {1, 2, 3, 4}.

R1 = {(2,2), (2,3),(2,4),(3,2),(3,3),(3,4)}

R2 = {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}

R3 = {2,4),(4,2)}

R4 = {(1,2),(2,3),(3,4)}

R5 = {(1,1),(2,2),(3,3),(4,4)}

a)  Which of these relations are reflexive? Justify your answers.

b)  Which of these relations are symmetric? Justify your answers.

c)  Which of these relations are antisymmetric? Justify your answer.

d)  Which of these relations are transitive? Justify your answers.

A. Provide formal of the validly of each of the following arguments.

1. I shall either play basketball or i shall go to the mall and watch the movie. therefore i shall either play basketball or go to a mall.
2. the Chocolate hills are not chocolate if I'm not the president of the Philippines. I'm not the president of the Philippines. therefore the Chocolate hills are chocolates.
3. if the summer is hot when we won't go on vocation in may. we'll either go on vacation inmay or we'll buy a boat (perhaps both). therefore, if the summer is hot, we'll buy a boat.
4. if Anna encore or write a program, she get an a. she's encoding and reading her lessons. therefore she'll get an A.

What is the solution for the recurrence relation a_n=2a_{n−1}−1

an

​=2an−1

​−1 with a_1=3

a1

​=3 .

C. Let p and q be proposition

p: 4 is a rational number

q: √3 is an irrational number

Express each of these proposition as an English sentences:

1. ~p
2. p∨ q
3. p ∧ q
4. p → q
5. p ↔ q