Discrete Mathematics Answers

Suppose that p and q are any statements. By constructing the truth tables, show that the

statement ¬ (p V q) & (¬ p) ∧ (¬ q) are logically equivalent.

Consider the functions from the set of students in a discrete mathematics class. Under what                      

             conditions is the function one-to-one if it assigns to a student his or her a) mobile phone    

             number. b) student identification number. c) final grade in the class. d) home town

Using a truth table, prove or disprove the following:

~[(~p∧q)↔ r]≡ ~(p∨~q)↔ ~r

Let A = {l, m, m, m, n, n, n, p, p, p} and B = {l, l, m, n, n, n, p, r, r}, find the operations of the following multisets:

A. A + B 

B.  A - B 

C. A U B

D.  A ∩ B

Let R ⊆ S × S be an equivalence relation on a set S. For an element x ∈ S, let

S(x) = {y ∈ S : (x, y) ∈ R}. Show that for every pair of elements x, y ∈ S, either

S(x) = S(y) or S(x) ∩ S(y) = ∅.

For each proposition below, decide whether it is true or false and give a brief explanation. Assume the universe (domain of variables) to be Z, the set of integers.

(1) ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))

(2) ∀x((x < 0) ∨ (x^2 ≥ x))

(3) ¬(∃xP(x)) ↔ (∀x¬P(x)) for all predicates P(x)

In how many ways can a set of five letters be selected from alphabets

Using the definition of "Big-O" determine if each of the following functions, f(x) = (xlogx)^2 - 4 and g(x) = 5x^5 are O(x^4) and prove your claims.

Determine for which positive integer values of n, 3n^3+2≤n^4 and prove your claim by mathematical induction.

I was able to determine n must be greater or equal to 4. But I can't seem to get through the inductive step.

Find the singleton sets and the pair of disjoint sets from the following

1. A = {x |x and x4 − 4 =0} and B = {y | y and x4 − 16 =0 =0}.