Discrete Mathematics Answers

The fibonacci sequence is defined as x0 = 0, x1 = 1 and xn+2 = xn +xn+1 , for all non negative integers n prove that, xm = xr+1xm-r + xrxm-r for all integers m ≥ 1 and 0 ≤ r ≤ m-1 and xd divides xkd for all integers k and d

Please read the following behavioral objective and identify the parts of the objectives as asked in the following questions.On the bus, Jonathan will sit quietly for 20 minutes, 80% of the time over 5 sessions.Choose the portion of the sentence that is the “what” condition.

A.
“will sit quietly”
B.
“on the bus”
C.
“Jonathan”
D.
“80% of the time over 5 sessions”

Chin-Tan found 52 action figures for his yard sale.
He wants to put them in more than 1 box, but fewer than 5 boxes.
Each box will have the same number of figures.
How many boxes can Chin-Tan use? Explain.

3. Stella Maris Schools, at one time, has 37 vacancies for teachers, out of which 22 are for English Language, 20 for History, 17 for Fine-Art. Of these vacancies, 11 are for both English Language and History, 8 for both, History and Fine Art, 7 for English Language and Fine-Art.
Using a Venn diagram or otherwise, find the number of teachers who must be able to teach:
a. All the three subjects.
b. Fine Art only
c. English Language and History not Fine Art.

Let A and B be finite sets for which |A|=|B| and suppose f: A —> B. Prove that f is injective if and only if f is surjective

b) Suppose that b is small and d is large, and further that either b and d are identical or that c is a
tetrahedron. Max believes that c must be a tetrahedron on the basis of this information. Which of
the following is true of this situation?

1. Max is correct to believe that we can determine the shape of c but it is not a tetrahedron,
since the given information is contradictory.
2. Max is mistaken, and nothing can be concluded about c, after all c is only mentioned in
one of the disjuncts of a disjunction.
3. Max is correct, because there is no situation in which b is small, d is large and b and d are
names of the same objects.
4. Max is correct, because it there is no situation in which b is small, d is large, and c is a
cube.

j) Suppose we translated Aristotle's famous sentence "All men are mortal" as ∀x(Man(x) ∧
Mortal(x)). Which domain could result in a counterexample to the claim that this is a good
translation of the English sentence.?

1. A collection of men, all of whom are mortal.
2. A collection of men all of whom are immortal
3. A collection of men, some mortal and some immortal.
4. A collection of mortal men, one aardvark and one penguin.

h) Which assumptions are being referred to by the phrase "the assumptions in force" at a given step
in a proof?

1. Only the premises at the top level of the proof.
2. Only the assumption of the subproof in which the step-in question is embedded.
3. All of the assumptions of every subproof in which the step-in question is embedded
together with the premises.
4. Only the assumptions of every subproof in which the step-in question is embedded.
5. Only the assumptions of subproofs of equally deep nesting.



i) Which of the following domains would most clearly suggests that the sentence " 'Everyone' is a
lot of people" is not always true?

1. The set of people in the province of Gauteng.
2. The set of logicians in this video correct
3. The set of even natural numbers
4. The set of people at a well-attended concert
5. The set of points on the interval [0,1)

f) Suppose we are told that the following expression is true: P ↔ (Q ∧ ¬Q). What can we then
conclude about P's truth?

1. P must be true
2. It is uncertain whether P is true or false
3. P must be false.


g) Which of the following informal arguments is the best example of a use of conditional elimination?
In addition to being able to recognize conditional elimination you will need to use your knowledge
of the correct translation of natural language conditionals.

1. “John will prove a theorem only if he isn't very tired. He slept very well last night, so he'll
prove a theorem."
2. “John won't prove theorems if he is tired. He's pretty tired today, so he'll prove a theorem."
3. ”If Dave doesn't sleep, then he'll prove a theorem. Since he slept soundly, he
consequently proved no theorems."
4. “Dave has a good night's sleep only if he will prove a theorem. Last night, he slept quite
well and so he will prove a theorem.

d) Assume ⊥ occurs in a subproof of the main proof. The appearance of ⊥ in this subproof indicates
that:

1. The premises of the main proof are mutually inconsistent.
2. The assumption of the subproof is a TT-contradiction.
3. One of the premises is the negation of the assumption of the subproof.
4. The premises of the main proof, together with the assumption of the subproof are
mutually inconsistent.


e) Recall that conjunction is idempotent, that is A ∧ A⇔A and commutative, that is A ∧ B⇔B ∧ A.
Which of these properties h old of material conditional (→).

1. Idempotent and commutative
2. Idempotent and not commutative
3. Commutative but not idempotent
4. Neither idempotent nor commutative