Question **34**

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Study the following arguments:

(i) Anne is smart if she knows discrete mathematics. She doesn't know discrete mathematics. Therefore, she is not smart.

(ii) All  parrots like  fruit. My  pet  bird is  not  a  parrot.  There­fore, my pet bird does not like  fruit. 

Then (i) is ... and (ii) is ...

Select one:

a.

logical, illogical

b.

illogical, logical

c.

illogical, illogical

d.

logical, logical

Clear my choice

Question **35**

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Given the adjacency matrix of an undirected graph

abc

a

2

1

3

b

1

1

1

c

3

1

0


How many **paths of length 3** are there from the vertex **b** to the vertex **a** in this graph?






Select one:

a.

33

b.

18

c.

27

d.

21

e.

None of these

Clear my choice

Question **36**

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Match a with b so that **a is congruent to b modulo 7**.


a = 17

Answer 1Choose...b = -3b = -5b = -4

a = 11

Answer 2Choose...b = -3b = -5b = -4

a = 23

Answer 3Choose...b = -3b = -5b = -4

Question **37**

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The function f(x) = x3 + 2x is ___

Select one:

a.

None of these

b.

O(x2)

c.

O(x3)

d.

O(3x)

Clear my choice

Question **38**

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Study the function f (k) satisfying

f(k) = 3f(k-1) - f(k-2), and *f* (1) = 1, f(2) = 2.

Find f(5).

Select one:

a.

15

b.

19

c.

None of the others

d.

18

e.

21

Clear my choice

Question **39**

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How many 1s are there in the adjacency matrix representing the graph K6?




Select one:

a.

15

b.

5

c.

30


d.

36

Clear my choice

Question **40**

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A full binary tree with 31 vertices has **height** at most ___


Select one:

a.

16

b.

None of tthese

c.

15

d.

30

e.

5

### Question 34

(i) Anne is smart if she knows discrete mathematics. She doesn't know discrete mathematics. Therefore, she is not smart.

This argument uses the logical form of **modus tollens**: \( P \rightarrow Q \) and \( \neg Q \) implies \( \neg P \).

(ii) All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit.

This argument is a logical fallacy called **denying the antecedent**: \( P \rightarrow Q \) and \( \neg P \) implies \( \neg Q \) (which is not necessarily true).

**Answer:** a. logical, illogical

### Question 35

To find the number of paths of length 3 from vertex \( b \) to vertex \( a \), we need to use the adjacency matrix of the graph and perform matrix exponentiation to find the number of walks of length 3. Given the adjacency matrix:

\[
\begin{matrix}
 & a & b & c \\
a & 2 & 1 & 3 \\
b & 1 & 1 & 1 \\
c & 3 & 1 & 0 \\
\end{matrix}
\]

We calculate the third power of the adjacency matrix and then check the element corresponding to \( b \) to \( a \).

**Answer:** b. 18

### Question 36

We need to find congruence relations modulo 7.

- \( 17 \equiv -4 \pmod{7} \)
- \( 11 \equiv -3 \pmod{7} \)
- \( 23 \equiv -5 \pmod{7} \)

**Answers:**

1. a = 17 is congruent to b = -4
2. a = 11 is congruent to b = -3
3. a = 23 is congruent to b = -5

### Question 37

The function \( f(x) = x^3 + 2x \) is asymptotically dominated by the highest degree term, which is \( x^3 \).

**Answer:** c. \( O(x^3) \)

### Question 38

The recurrence relation is given by:
\[
f(k) = 3f(k-1) - f(k-2)
\]

With initial conditions:
\[
f(1) = 1, \quad f(2) = 2
\]

Calculate subsequent values:
\[
\begin{aligned}
f(3) &= 3f(2) - f(1) = 3(2) - 1 = 5 \\
f(4) &= 3f(3) - f(2) = 3(5) - 2 = 13 \\
f(5) &= 3f(4) - f(3) = 3(13) - 5 = 34 - 5 = 29 \\
\end{aligned}
\]

**Answer:** c. None of the others

### Question 39

The complete graph \( K_6 \) has an adjacency matrix where every off-diagonal entry is 1 and diagonal entries are 0. For a \( K_6 \):

\[
\text{Total entries} = 6 \times 6 = 36
\]
\[
\text{Diagonal entries (0s)} = 6
\]
\[
\text{Number of 1s} = 36 - 6 = 30
\]

**Answer:** c. 30

### Question 40

For a full binary tree with 31 vertices, the height \( h \) of the tree is given by:
\[
n = 2^{h+1} - 1
\]

Solving for \( h \) when \( n = 31 \):
\[
31 = 2^{h+1} - 1 \implies 2^{h+1} = 32 \implies h+1 = 5 \implies h = 4
\]

**Answer:** b. None of these (as the correct height is 4)

---

Would you like more details or have any questions?

Here are five more questions you might be interested in:

1. How to solve recurrence relations?
2. How to find the order of growth of functions in Big-O notation?
3. What is the logical form of modus ponens and modus tollens?
4. How to compute powers of an adjacency matrix?
5. What are the properties of congruence relations?

**Tip:** When solving problems involving matrix exponentiation, consider using software tools for matrix calculations to avoid errors and save time.

Explore logical arguments like Modus Tollens and logical fallacies like Denying the Antecedent. Solve graph theory problems using adjacency matrices and matrix exponentiation. Suitable for advanced undergraduate students.

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