in exercises 15–22, write the conditional statement ( p o q ), the converse ( q o p ), the inverse (
eg p o
eg q ), and the contrapositive (
eg q o
eg p ) in words. then decide whether each statement is true or false. (see example 1.)
15. let ( p ) be “two angles are supplementary” and let ( q ) be “the measures of the angles sum to ( 180^circ ).”
16. let ( p ) be “you are in math class” and let ( q ) be “you are in geometry.”
17. let ( p ) be “you lift weights” and let ( q ) be “you exercise.”
18. let ( p ) be “you are sick with a cold” and let ( q ) be “you have a stuffy nose.” (handwritten note: rue, ...\)
19. let ( p ) be “an instrument is a cuatro” and let ( q ) be “the instrument is a guitar.”
20. let ( p ) be “the sun is out” and let ( q ) be “it is daytime.”
21. let ( p ) be ( 3x - 7 = 20 ) and let ( q ) be ( x = 9 ).
22. let ( p ) be “it is pascua florida day” and let ( q ) be “it is april.”

Question

in exercises 15–22, write the conditional statement ( p 	o q ), the converse ( q 	o p ), the inverse (
eg p 	o
eg q ), and the contrapositive (
eg q 	o
eg p ) in words. then decide whether each statement is true or false. (see example 1.)
15. let ( p ) be “two angles are supplementary” and let ( q ) be “the measures of the angles sum to ( 180^circ ).”
16. let ( p ) be “you are in math class” and let ( q ) be “you are in geometry.”
17. let ( p ) be “you lift weights” and let ( q ) be “you exercise.”
18. let ( p ) be “you are sick with a cold” and let ( q ) be “you have a stuffy nose.” (handwritten note: 	rue, ...\)
19. let ( p ) be “an instrument is a cuatro” and let ( q ) be “the instrument is a guitar.”
20. let ( p ) be “the sun is out” and let ( q ) be “it is daytime.”
21. let ( p ) be ( 3x - 7 = 20 ) and let ( q ) be ( x = 9 ).
22. let ( p ) be “it is pascua florida day” and let ( q ) be “it is april.”

Accepted Answer

Let's solve problem 15 first (as it's a clear one with angle - related logic, and we can follow the steps for conditional statements):

### Problem 15:
Given $ p $: "Two angles are supplementary" and $ q $: "the measures of the angles sum to $ 180^{\circ} $"

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate the logical implication $ p
ightarrow q $ into words. The conditional statement means if $ p $ is true, then $ q $ is true.
So, "If two angles are supplementary, then the measures of the angles sum to $ 180^{\circ} $".
We know that by definition, supplementary angles are two angles whose measures add up to $ 180^{\circ} $. So this statement is True.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: The converse of $ p
ightarrow q $ is $ q
ightarrow p $, which means we reverse the order of $ p $ and $ q $ in the implication.
So, "If the measures of two angles sum to $ 180^{\circ} $, then the two angles are supplementary".
By the definition of supplementary angles, this is also True (since that's exactly what supplementary angles are defined as).

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: The inverse of a conditional statement $ p
ightarrow q $ is formed by negating both $ p $ and $ q $. The negation of $ p $ ("Two angles are supplementary") is "Two angles are not supplementary", and the negation of $ q $ ("the measures of the angles sum to $ 180^{\circ} $") is "the measures of the angles do not sum to $ 180^{\circ} $".
So, "If two angles are not supplementary, then the measures of the angles do not sum to $ 180^{\circ} $".
If two angles are not supplementary, their measures do not add up to $ 180^{\circ} $, so this statement is True.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: The contrapositive of $ p
ightarrow q $ is formed by negating both $ p $ and $ q $ and reversing their order. The negation of $ q $ is "the measures of the angles do not sum to $ 180^{\circ} $" and the negation of $ p $ is "two angles are not supplementary".
So, "If the measures of two angles do not sum to $ 180^{\circ} $, then the two angles are not supplementary".
If the sum of two angles is not $ 180^{\circ} $, they can't be supplementary (by definition), so this statement is True.

### Problem 16:
Given $ p $: "you are in math class" and $ q $: "you are in Geometry"

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words. It means if the first statement is true, the second is true.
So, "If you are in math class, then you are in Geometry".
This is False because there are other math classes (like Algebra, Calculus, etc.) besides Geometry.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If you are in Geometry, then you are in math class".
This is True because Geometry is a type of math class.

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $ and form the implication. The negation of $ p $ is "you are not in math class" and the negation of $ q $ is "you are not in Geometry".
So, "If you are not in math class, then you are not in Geometry".
This is True. If you are not in any math class, you can't be in Geometry (which is a math class).

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is "you are not in Geometry" and the negation of $ p $ is "you are not in math class".
So, "If you are not in Geometry, then you are not in math class".
This is False. You could be in another math class (like Algebra) even if you are not in Geometry.

### Problem 17:
Given $ p $: "you lift weights" and $ q $: "you exercise"

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words.
So, "If you lift weights, then you exercise".
This is True because lifting weights is a form of exercise.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If you exercise, then you lift weights".
This is False. There are other forms of exercise (like running, swimming, etc.) that don't involve lifting weights.

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $ and form the implication. The negation of $ p $ is "you do not lift weights" and the negation of $ q $ is "you do not exercise".
So, "If you do not lift weights, then you do not exercise".
This is False. You could exercise in other ways (like running) even if you don't lift weights.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is "you do not exercise" and the negation of $ p $ is "you do not lift weights".
So, "If you do not exercise, then you do not lift weights".
This is True. If you don't exercise at all, you can't be lifting weights (since lifting weights is a type of exercise).

### Problem 18:
Given $ p $: "you are sick with a cold" and $ q $: "you have a sublime" (I think this might be a typo, maybe "you have a stuffy nose" or similar, but assuming $ p $ and $ q $ as given)

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words.
So, "If you are sick with a cold, then you have a sublime" (assuming "sublime" is a symptom, but if it's a typo, the logic would change. But based on the given, if we take it as is). If having a "sublime" is a symptom of a cold, then this could be True, but if "sublime" is not related, it's False. But likely a typo, maybe "stuffy nose". Let's assume a common cold symptom: If you are sick with a cold, then you have a stuffy nose (adjusting for the typo). Then this could be True (but not all colds have a stuffy nose, but for the sake of the problem).

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If you have a sublime (or stuffy nose), then you are sick with a cold".
This is False. A stuffy nose can be caused by allergies, sinus infections, etc., not just a cold.

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $ and form the implication. The negation of $ p $ is "you are not sick with a cold" and the negation of $ q $ is "you do not have a sublime (or stuffy nose)".
So, "If you are not sick with a cold, then you do not have a sublime (or stuffy nose)".
This is False. You could have a stuffy nose due to allergies even if you don't have a cold.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is "you do not have a sublime (or stuffy nose)" and the negation of $ p $ is "you are not sick with a cold".
So, "If you do not have a sublime (or stuffy nose), then you are not sick with a cold".
This is True (assuming that a cold always has a stuffy nose, which is not entirely accurate in real - life, but for the problem's logic).

### Problem 19:
Given $ p $: "an instrument is a cuatro" and $ q $: "the instrument is a guitar"

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words.
So, "If an instrument is a cuatro, then the instrument is a guitar".
A cuatro is a type of guitar - like instrument (in the broad sense of string -ed instruments in some classifications, but more precisely, a cuatro is a distinct instrument but related to the guitar family). So this could be considered True (depending on the classification). But in a strict sense, a cuatro is not the same as a guitar, so it could be False. Let's assume the problem's context: If we consider a cuatro as a type of guitar, then True; else, False.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If an instrument is a guitar, then the instrument is a cuatro".
This is False. There are many types of guitars (acoustic, electric, bass, etc.) that are not cuatros.

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $ and form the implication. The negation of $ p $ is "an instrument is not a cuatro" and the negation of $ q $ is "the instrument is not a guitar".
So, "If an instrument is not a cuatro, then the instrument is not a guitar".
This is False. An instrument could be a guitar (like an acoustic guitar) and not a cuatro.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is "the instrument is not a guitar" and the negation of $ p $ is "an instrument is not a cuatro".
So, "If an instrument is not a guitar, then the instrument is not a cuatro".
This is True. If an instrument is not a guitar, it can't be a cuatro (if we consider a cuatro as a type of guitar).

### Problem 20:
Given $ p $: "the sun is out" and $ q $: "it is daytime"

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words.
So, "If the sun is out, then it is daytime".
This is True because the sun is out during the daytime.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If it is daytime, then the sun is out".
This is False. There can be cloudy days or overcast days during the daytime when the sun is not visible (but it's still daytime).

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $ and form the implication. The negation of $ p $ is "the sun is not out" and the negation of $ q $ is "it is not daytime".
So, "If the sun is not out, then it is not daytime".
This is False. It can be daytime (e.g., cloudy day) even if the sun is not out.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is "it is not daytime" and the negation of $ p $ is "the sun is not out".
So, "If it is not daytime, then the sun is not out".
This is True. If it's not daytime (e.g., night), the sun is not out.

### Problem 21:
Given $ p $: " $ 3x - 7=20 $" and $ q $: " $ x = 9 $"

First, solve $ p $:
$ 3x-7 = 20 $
Add 7 to both sides: $ 3x=20 + 7=27 $
Divide both sides by 3: $ x = 9 $

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words.
So, "If $ 3x - 7=20 $, then $ x = 9 $".
Since when we solve $ 3x - 7 = 20 $, we get $ x = 9 $, this statement is True.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If $ x = 9 $, then $ 3x - 7=20 $".
Substitute $ x = 9 $ into $ 3x-7 $: $ 3\times9 - 7=27 - 7 = 20 $. So this statement is True.

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $. The negation of $ p $ is " $ 3x - 7
eq20 $" and the negation of $ q $ is " $ x
eq9 $". Then form the implication.
So, "If $ 3x - 7
eq20 $, then $ x
eq9 $".
If $ 3x - 7
eq20 $, then $ 3x
eq27 $, so $ x
eq9 $. This statement is True.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is " $ x
eq9 $" and the negation of $ p $ is " $ 3x - 7
eq20 $".
So, "If $ x
eq9 $, then $ 3x - 7
eq20 $".
If $ x
eq9 $, then $ 3x
eq27 $, so $ 3x - 7
eq20 $. This statement is True.

### Problem 22:
Given $ p $: "it is Pascua Florida Day" and $ q $: "it is April"

Pascua Florida Day is on April 2nd (in Florida, USA).

#### Step 1: Conditional Statement ($ p
ightarrow q $)
**Explanation**: Translate $ p
ightarrow q $ to words.
So, "If it is Pascua Florida Day, then it is April".
Since Pascua Florida Day is in April, this statement is True.

#### Step 2: Converse ($ q
ightarrow p $)
**Explanation**: Reverse $ p $ and $ q $ in the implication.
So, "If it is April, then it is Pascua Florida Day".
This is False. April has 30 days, and Pascua Florida Day is only on one day (April 2nd) in April.

#### Step 3: Inverse ($
eg p
ightarrow
eg q $)
**Explanation**: Negate both $ p $ and $ q $. The negation of $ p $ is "it is not Pascua Florida Day" and the negation of $ q $ is "it is not April". Then form the implication.
So, "If it is not Pascua Florida Day, then it is not April".
This is False. It could be another day in April (not Pascua Florida Day) and still be April.

#### Step 4: Contrapositive ($
eg q
ightarrow
eg p $)
**Explanation**: Negate both $ p $ and $ q $ and reverse their order. The negation of $ q $ is "it is not April" and the negation of $ p $ is "it is not Pascua Florida Day".
So, "If it is not April, then it is not Pascua Florida Day".
This is True. If it's not April, Pascua Florida Day (which is in April) can't occur.

### Final Answers (for each problem's statements):

#### Problem 15:
- Conditional ($ p
ightarrow q $): True
- Converse ($ q
ightarrow p $): True
- In

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 15:

Step 1: Conditional Statement (\( p

Step 2: Converse (\( q

Step 3: Inverse (\(

Step 4: Contrapositive (\(

Answer:

Final Answers (for each problem's statements):

Problem 15: