Discrete Mathematics Answers

Construct a directed graph for the board members of the company if the President can
influence the Director of research and Development, the Director of Marketing,
and the Director of Operations; the Director of Research and Development can
influence the Director of Operations; the Director of Marketing can influence the
Director of Operations; and no one can influence, or be influenced by, the Chief
Financial Officer.

At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether?

Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation R from A to B as follows: For all (x, y)∈ A ×B,
(x, y)∈ R means that is an integer.
a. Is 4 R 6? Is 4 R 8? Is (3, 8) ∈R? Is (2, 10) ∈R?

What reads "the goods are are standard if and only if the goods are expensive"?

which of the following statement is true

a. \\((p\\wedge q)\\vee (p\\vee r)=p\\vee (q\\wedge r)\\)
b. \\((p\\vee q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\\)
c. \\(\\sim (p\\vee q)=\\sim (p\\wedge \\sim q)\\)
d. \\((p\\wedge q)\\wedge (p\\vee r)=p\\vee (q\\wedge r)\\)

Show that 1n^3+2n+3n^2 is divided by 2 and 3 for all positive integers n

Use the principle of induction Prove that 2 1 n 2 n 2
n n 1
> + ∀ >
−
,

Express the negation of the following statements WITHOUT using the negation symbol:
a. ∀x(−2 < x <3)
b. ∀x(0 ≤ x <5)
c. ∃x(−4 ≤ x ≤1)
d. ∃x(−5 < x <−1)
e. ∀x∃y(x2 < y)

Suppose g : A → B and f : B → C are functions.
a. Show that if f ◦g is onto, then f must also be onto.
b. Show that if f ◦g is one-to-one, then g must also be one-to-one.
c. Show that if f ◦g is a bijection, then g is onto if and only if f is one-to-one.