Discrete Mathematics Answers

The famous detective TVTHREE, Kara Singh Walla was called in to solve a baffling
murder mystery. He determined the following facts:

A. Alan, the murdered man, was killed by a blow on the head with brass
candlestick.

B. Either Alan’s wife or a maid, Sara, was in the dining room at the time of the
murder.

C. If the cook was in the kitchen at the time of the murder, then the butler killed
Alan with fatal dose of strychnine.

D. If the Alan’s wife was in the dining room at the time of the murder, then the
chauffeur killed Alan.

E. If the cook was not in the kitchen at the time of the murder, then Sara was not
in the dining room when the murder committed.

F. If Sara was in the dining room at the time the murder was committed, then the
wine steward killed Alan.

a. Write each primary statement above using the variables p, q, r, s, t, and u.
b. Write all the hypotheses or premises based on the statements given.

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)