Discrete Mathematics Answers

I. Create a truth table for all of the 11 Logical Equivalence.

II. Find if the following are logically equivalent

   1.(~p v q) ^ (~q) <=> ~(p v q)

   2.(p ^ ~q) v (~p v q) <=> T

Draw Hasse diagram repressing the partial order on {(a,b) : a\b } on {1,2,3,4,6,8,12}

illustrate the venn diagram for the sets A,B,and C in the U={1,2,3,4,5,6,7,8} a={1,2,3,4,5,6,7}, b={1,5,6,7}, c={1,2,3,6}

Suppose that the domain of the predicateP(x) consists of −2, −1, 0, 1, and2. Write out each

of the following predicate logic formulas in propositional logic formulas using disjunctions, conjunctions,

negations, or their combinations.

Write the negation of the following statement:

∀𝑥∃𝑦(𝑥 + 𝑦 = 2 ∧ 2𝑥 − 𝑦 = 1

Solve the following recurrence relations for the initial conditions given. The population of Utopia increases 5 percent per year. In 2000 the population was 10,000. What was the population in 1970?

P = I pass this class

G = I go to class every day.

H = I do all the homework exercises.

Translate the following sentences into propositional logic.

(1) Students will not pass this class unless they go to class every day and do all of the 

homework exercises.

(2) Either going to classes every day or doing all the homework exercises is necessary 

for passing this class for students.

(3) There is no student in the class who goes to classes every day and does all the 

homework exercises but will not pass this class.(4) To pass in the class it is necessary and sufficient to go to classes every day or do all 

the homework exercises.

(5) Either go to classes every day or do all the homework exercises but both is not 

required if students wants to pass the

1. Expand the following using the Binomial Theorem

a. (x-3y)^3

b. (2a+b^-3)^4

c. (a+b+c)²

2. Solve the recurrence relation Pn = Pn-1 + n with the initial condition p_1 = 2 using iteration.

Find the domain and range of these functions. Note that in each case, to find the domain, determine the set of elements assigned values by the function. a) the function that assigns to each bit string the number of ones in the string minus the number of zeros in the string b) the function that assigns to each bit string twice the number of zeros in that string c) the function that assigns the number of bits left over when a bit string is split into bytes (which are blocks of 8 bits) d) the function that assigns to each positive integer the largest perfect square not exceeding this integer 

Write A= {10, 20, 30, 40, 50} using rule method.