Discrete Mathematics Answers

3. Let P(x), Q(x), and R(x) be the statements “x is a professor,” “x is ignorant,” and “x is vain,” respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), and R(x), where the domain consists of all people

a) No professors are ignorant.

b) All ignorant people are vain.

c) No professors are vain.

d) Does (c) follow from (a) and (b)?

2. Equal sets are always equivalent but equivalent sets may not be equal. Justify

A function f is said to be one-to-one, or an injection, if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. Note that a function f is one-to-one if and only if f(a) ≠ f(b) whenever a ≠ b. This way of expressing that f is one-to-one is obtained by taking the contrapositive of the implication in the definition. A function f from A to B is called onto, or a surjection, if and only if for every element b ∈ B there is an element a ∈ A with f(a) = b. A function f is onto if ∀y∃x( f(x) = y), where the domain for x is the domain of the function and the domain for y is the codomain of the function

Now consider that f is a function from A to B, where A and B are finite sets with |A| = |B|. Show that f is one-to-one if and only if it is onto.

. 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

(𝑝 → 𝑞) ↔ (𝑞 ∨ ~𝑝)

~(pVq)→r

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

2.

1. Use iteration, either forward or backward substitution, to solve the recurrence relation an=an−1−1 for any positive integer n, with initial condition, a0=1. Use mathematical induction to prove the solution you find is correct.

2. Determine the cardinality of each of the sets, A, B, and C, defined below, and prove the cardinality of any set that you claim is countably infinite.

A is the set of negative odd integers

B is the set of positive integers less than 1000

C is the set of positive rational numbers with numerator equal to 1.

3.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.

Let p and q be the propositions

p : It is below freezing. (including negations).

q : It is snowing

Write these propositions using p and q and logical connectives

a) It is below freezing and snowing.

b) It is below freezing but not snowing.

c) It is not below freezing and it is not snowing.

d) It is either snowing or below freezing (or both).

e) If it is below freezing, it is also snowing.

f ) Either it is below freezing or it is snowing, but it is

not snowing if it is below freezing.

Find the inverse of 35 modulo 11 by using extended Euclidean Algorithm

step by step solution