Discrete Mathematics Answers

When two sets A and B consist of the same elements, they are called_____sets
a.equal
b.compliment
c.union
d.difference

in a survey of 100 students, 56 wrote the Maths exams, 23 wrote psychology and 21 wrote the science exam. 12 wrote both maths and psychology exams, 9 write the maths and science exams and 6 wrote both psychology and science exams. 5 students wrote neither. determine how many students wrote all three exams.

Search Results
Web results

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example. (1) There are at least two ways of describing the set {7, 8....}. (2) Any function with domain R®R is a binary operation.

A is sufficient for B’ is equivalent to ‘the negative of A is necessary for the
negative of B’. is it true or false?

Obtain the order of each element of S(℘) where S={1,2,3}

4.2: For the proposition pairs below, create a truth table and compare each proposition’s truth profile: determine whether the pair is logically equivalent, contradictory, consistent or inconsistent.

Example: (¬J ≡ K) with [(J → ¬K) ∧ (¬K → J)]

Write your answer as follows:
Step 1
J K (¬J ≡ K) [(J → ¬K) ∧ (¬K → J)]
T T (F) ≡ T (T → F) ∧ (F → T)
T F (F) ≡ F (T → T) ∧ (T → T)
F T (T) ≡ T (F → F) ∧ (F → F)
F F (T) ≡ F (F → T) ∧ (T → F)

Step 2
J K (¬J ≡ K) [(J → ¬K) ∧ (¬K → J)]
T T F (F) ∧ (T)
T F T (T) ∧ (T)
F T T (T) ∧ (T)
F F F (T) ∧ (F)

Step 3
J K (¬J ≡ K) [(J → ¬K) ∧ (¬K → J)]
T T F F
T F T T
F T T T
F F F F

Answer: (¬J ≡ K) and [(J → ¬K) ∧ (¬K → J)] are logically equivalent


34. (P ∧ Q) ∨ P with (P ∨ Q) ∧ Q


35. [(S → W) → X] with [(S ∧ W) ∨ (W ∧ X)]

QUESTION 5

In this question you have to construct formal proofs using the natural deduction rules. The Fitch system makes use of these rules.

Remember that De Morgan’s laws and other tautologies are not permissible natural deduction rules. You are also not allowed to use Taut Con, Ana Con or FO Con. It is important to
number your statements, to indicate subproofs and at each step to give the rule that you are using

Question 5.1

Using the natural deduction rules, prove that the following two premises are contradictory:

1 R → (P v Q)
2 R /\ ¬P /\ ¬Q


Question 5.2

Using the natural deduction rules, give a formal proof of

| 1. ¬S
| 2. P → Q
| 3. Q → (R v S)
| 4. P v R
|---
| R


Question 5.3

Prove ∃x[P(x) → ∀yP(y)] from no premises

QUESTION 3

Consider the arguments below and decide whether they are valid. If they are, write down an informal proof, phrased in complete, well-formed English sentences. If the argument is invalid, construct a counter example.

In Questions 3.1 and 3.3 we assume we deal with the blocks language.

Question 3.1

| x or y is at home but either v or z is unhappy
| Either x is not home or z is happy.
| Either y is not home or z is unhappy
|---
| z is unhappy


Question 3.2

| Aggie or Cecil is not shopping.
| Cecil is shopping or Cecil and Aggie are married.
| Aggie and Cecil are not married or Aggie is shopping.
|----
| Cecil and Aggie are married.


Question 3.3

| Student(peter) ∨ Hungry(peter, 2:00)
| ¬ Hungry(peter, 2:00) ∨ ¬Pet(patience)
| Pet(patience) ∨ ¬ Student(peter)
|----
| Student(peter)

QUESTION 2

Which of the following statements motivate the use of informal proof? Answer true or false to the
following informal proof statements:

a) Truth tables cannot demonstrate logical consequence for formulas containing more than 10
atoms.

b) Informal proof can sometimes explain why a logical consequence holds better than alternative
methods.

c) Not all connectives are truth functional, and hence truth tables do not capture the fully general
case of logical consequence.

d) Informal proofs contain no symbols and so can be understood by everyone.

QUESTION 4

Below are a number of expressions. State which are terms, some are atomic wffs (well-formed formulae) and some are neither.

a) Tet(y)

b) Logician(john)

c) father_of(quinn)

d) 2 + y = z2

e) Angry(x; y;2:00)