Discrete Mathematics Answers

Prove by mathematical induction. 1 + 5 + 9 +...+ (4n-3) = (2n+1)(n-1) for n ≥ 2

Find the number of positive integers less than 601 that are not divisible by 4 or 5 or 6

In a company, there are 50 employees and some committees. If each employee belongs

to 6 committees and each committee consists of 10 people, how many committees are there?

Find a common domain for the variables x, y, z, and w for which the statement ∀x∀y∀z∃w((w ≠ x) ∧ (w ≠ y) ∧ (w ≠ z)) is true and another common domain for these variables for which it is false

. 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

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)