decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients to
make decisions about their own bodies and medical treatments. This principle is enshrined in the legal
concept of informed consent. Informed consent requires healthcare providers to disclose all relevant
information about the risks, benefits, and alternatives to a medical treatment or procedure, allowing
patients to make informed decisions.However, challenges arise when patients are not fully capable of
making informed decisions (e.g., due to age, mental illness, or language barriers). In such cases, ethic
Chapter 2 – Reasoning and ProofAnswer Key
2.1 Conjectures and Counterexamples
Answers
1. True
2. COULD NOT IDENTIFY THE CONJECTURE
3. False, maybe a raccoon ate the bread with peanut butter instead of the chipmunk
4. False, it is possible that the other students, that are not her friends, in her geometry class are
at different grade levels.
5. Answers vary
6. n = 1 would be a counterexample because 12 . Recall that a whole number is
0, 1, 2, 3, … n, so 0 could also be a counterexample.
7. Counterexamples include: 21, 51, 81, 121, and 151
, 8. is one counterexample. Any positive improper fraction (where the numerator is greater
than the denominator) could be a counterexample.
9. A triangle is a counterexample.
10. A girl that doesn’t like ice cream would be a counterexample.
decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients
to make decisions about their own bodies and medical treatments. This principle is enshrined in
the legal concept of informed consent. Informed consent requires healthcare providers to disclose
all relevant information about the risks, benefits, and alternatives to a medical treatment or
procedure, allowing patients to make informed decisions.However, challenges arise when patients
are not fully capable of making informed decisions (e.g., due to age, mental illness, or language
barriers). In such cases, ethical
11. Not everyone takes choir in high school.
12. Obtuse angles do not have complementary angles.
13. 13, 14, 15 year-olds are teenagers and cannot drive yet.
14. Any negative integer could be a counterexample.
15. Equations can have any real number as a solution.
, 2.2 If-Then Statements
Answers
1. Hypothesis: 5 divides evenly into x.
Conclusion: x ends in 0 or 5.
2. Hypothesis: A triangle has three congruent sides.
Conclusion: It is an equilateral triangle.
3. Here, the ―if‖ is in the middle of the statement, making the hypothesis the second half.
Hypothesis: Three points lie in the same plane.
Conclusion: The three points are coplanar.
4. Hypothesis: x = 3.
Conclusion: .
decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients to ma
decisions about their own bodies and medical treatments. This principle is enshrined in the legal conce
of informed consent. Informed consent requires healthcare providers to disclose all relevant information
about the risks, benefits, and alternatives to a medical treatment or procedure, allowing patients to mak
informed decisions.However, challenges arise when patients are not fully capable of making informed
decisions (e.g., due to age, mental illness, or language barriers). In such cases, ethical
5. Hypothesis: You take yoga.
Conclusion: You are relaxed.
6. Hypothesis: You are a baseball player.
Conclusion: You wear a hat.
7. Hypothesis: I am 16 years old.
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients to
make decisions about their own bodies and medical treatments. This principle is enshrined in the legal
concept of informed consent. Informed consent requires healthcare providers to disclose all relevant
information about the risks, benefits, and alternatives to a medical treatment or procedure, allowing
patients to make informed decisions.However, challenges arise when patients are not fully capable of
making informed decisions (e.g., due to age, mental illness, or language barriers). In such cases, ethic
Chapter 2 – Reasoning and ProofAnswer Key
2.1 Conjectures and Counterexamples
Answers
1. True
2. COULD NOT IDENTIFY THE CONJECTURE
3. False, maybe a raccoon ate the bread with peanut butter instead of the chipmunk
4. False, it is possible that the other students, that are not her friends, in her geometry class are
at different grade levels.
5. Answers vary
6. n = 1 would be a counterexample because 12 . Recall that a whole number is
0, 1, 2, 3, … n, so 0 could also be a counterexample.
7. Counterexamples include: 21, 51, 81, 121, and 151
, 8. is one counterexample. Any positive improper fraction (where the numerator is greater
than the denominator) could be a counterexample.
9. A triangle is a counterexample.
10. A girl that doesn’t like ice cream would be a counterexample.
decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients
to make decisions about their own bodies and medical treatments. This principle is enshrined in
the legal concept of informed consent. Informed consent requires healthcare providers to disclose
all relevant information about the risks, benefits, and alternatives to a medical treatment or
procedure, allowing patients to make informed decisions.However, challenges arise when patients
are not fully capable of making informed decisions (e.g., due to age, mental illness, or language
barriers). In such cases, ethical
11. Not everyone takes choir in high school.
12. Obtuse angles do not have complementary angles.
13. 13, 14, 15 year-olds are teenagers and cannot drive yet.
14. Any negative integer could be a counterexample.
15. Equations can have any real number as a solution.
, 2.2 If-Then Statements
Answers
1. Hypothesis: 5 divides evenly into x.
Conclusion: x ends in 0 or 5.
2. Hypothesis: A triangle has three congruent sides.
Conclusion: It is an equilateral triangle.
3. Here, the ―if‖ is in the middle of the statement, making the hypothesis the second half.
Hypothesis: Three points lie in the same plane.
Conclusion: The three points are coplanar.
4. Hypothesis: x = 3.
Conclusion: .
decisions that not only have life-altering consequences for patients but also raise profound moral
questions. These issues include:#### 2.1 **Patient Autonomy and Informed Consent**One of the
fundamental ethical principles in healthcare is respect for patient autonomy—the right of patients to ma
decisions about their own bodies and medical treatments. This principle is enshrined in the legal conce
of informed consent. Informed consent requires healthcare providers to disclose all relevant information
about the risks, benefits, and alternatives to a medical treatment or procedure, allowing patients to mak
informed decisions.However, challenges arise when patients are not fully capable of making informed
decisions (e.g., due to age, mental illness, or language barriers). In such cases, ethical
5. Hypothesis: You take yoga.
Conclusion: You are relaxed.
6. Hypothesis: You are a baseball player.
Conclusion: You wear a hat.
7. Hypothesis: I am 16 years old.
