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.
illogical, illogical

b.
logical, illogical

c.
logical, logical

d.
illogical, logical

Question 28
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Which proposition is logically equivalent to (p → r) ^ (¬p → r)?

(i) r

(ii) p v ¬p v r

(iii) ¬r

(iv) p ^ r

Select one:

a.
(i)

b.
(iii)

c.
(iv)

d.
(ii)

Question 29
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Construct a binary search tree for the sentence

"Do not count your chickens before they are hatched"

then find how many comparisons are needed to search the word "hatched".
Select one:

a.
3

b.
2

c.
4

d.
5

Question 30
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Given f(x) = x + 2  and g(x) = x3.

What is the composite function (fog)(x)?

Select one:

a.
(fog)(x) = x3 + 2

b.
(fog)(x) = x3(x+2)

c.
None of the others

d.
(fog)(x) = (x+2)3

Let's analyze each question one by one:

### Question 1

**Argument Analysis:**

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

This argument can be written as:
- Premise: If Anne knows discrete mathematics, then Anne is smart.
- Premise: Anne does not know discrete mathematics.
- Conclusion: Therefore, Anne is not smart.

This is a logical fallacy known as "denying the antecedent." The conclusion does not logically follow from the premises.

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

This argument can be written as:
- Premise: All parrots like fruit.
- Premise: My pet bird is not a parrot.
- Conclusion: Therefore, my pet bird does not like fruit.

This is a logical fallacy known as "denying the antecedent." Just because all parrots like fruit does not mean that only parrots like fruit.

Thus, (i) is illogical and (ii) is illogical.

The correct choice is:
**a. illogical, illogical**

### Question 2

Which proposition is logically equivalent to \((p \rightarrow r) \land (\neg p \rightarrow r)\)?

Let's break down the propositions:

- \(p \rightarrow r\) is equivalent to \(\neg p \lor r\)
- \(\neg p \rightarrow r\) is equivalent to \(p \lor r\)

So, \((p \rightarrow r) \land (\neg p \rightarrow r)\) simplifies to:
\[(\neg p \lor r) \land (p \lor r)\]

This means \(r\) must be true regardless of the truth values of \(p\). Hence, the equivalent proposition is simply \(r\).

The correct choice is:
**a. (i)**

### Question 3

Construct a binary search tree for the sentence "Do not count your chickens before they are hatched," and find how many comparisons are needed to search for the word "hatched."

The binary search tree for the words in alphabetical order would be:

```
          Do
           \
           not
          /   \
      count   your
     /    \      \
 chickens  before they
             /       \
            are     hatched
```

To search for "hatched":

1. Compare with "Do" (hatched > Do)
2. Compare with "not" (hatched < not)
3. Compare with "count" (hatched > count)
4. Compare with "before" (hatched > before)
5. Compare with "they" (hatched < they)
6. Compare with "hatched" (found)

So, 5 comparisons are needed.

The correct choice is:
**d. 5**

### Question 4

Given \(f(x) = x + 2\) and \(g(x) = x^3\), what is the composite function \((f \circ g)(x)\)?

\[(f \circ g)(x) = f(g(x))\]

Given \(g(x) = x^3\), then:

\[f(g(x)) = f(x^3) = x^3 + 2\]

The correct choice is:
**a. (fog)(x) = x^3 + 2**

---

Do you want details or have any questions?

Here are 5 questions you can ask:
1. How do logical fallacies impact argument validity?
2. Can you explain the concept of composite functions with more examples?
3. How is a binary search tree constructed from a list of words?
4. What are common logical equivalences in propositional logic?
5. Can you provide more examples of logical arguments and their validity?

**Tip:** When constructing binary search trees, ensure each comparison helps to place elements correctly, maintaining the tree's order properties.

Explore logical arguments, logical equivalences, binary search trees, and composite functions in mathematics with detailed examples and solutions. Perfect for college-level students studying mathematical logic and algorithms.

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