Review Notes for Term Test 1, MAT 137
Maki Lu
October 11, 2023
1 Introduction
1.1 Purpose of the Review Notes
I am Maki, a 2nd year math PhD student and a TA in this course for TUT0402
and TUT0803. Some of my students from my tutorials prepared some review
notes and want to share with everyone in the course, so I am glad to type it
down using LaTeX and add some important tips so that you can take advantage
of this file for the term test.
1.2 Reference
The reference is my students’ summaries of what they have learned from the
lectures, tutorials, textbooks, videos, practice problems and some past problem
sets. This may not be a full list, but it should be helpful in some sense.
I am proud of my students, and I am proud to share these notes to all of
you, and I want these notes to be helpful. If you find any typos, you can email
me.
2 How to write proofs?
In each subsection, we give the instructions to prove statements with examples.
2.1 Simple Statements
2.1.1 p
To show p directly, we might do some calculations, and in the end deduce p.
Exercise 1. Show: 32 + 3 > 10.
Proof. Note that 32 + 3 = 12 > 10, so the statement is true.
1
, 2.1.2 p and q
If we want to show p and q, it breaks into two questions: one is to show p is
true, and the other is to show q is true.
Exercise 2. Show: 32 + 3 > 10 and 3 > 1.
Proof.
1. First, 32 + 3 = 12 > 10.
2. Also, 3 is clearly bigger than 1.
This tells us that the statement is true.
2.1.3 p or q
If we want to show p or q, it suffices to show that at least one of them is true.
Exercise 3. Show: 1 > 3 or 3 > 1.
Proof. Clearly, 3 > 1. The second statement is true. So the entire statement is
true.
2.1.4 p =⇒ q
If we want to show p implies q, it suffices to first assume that p is true, and use
the assumption of p to prove that q is true.
Exercise 4. Show: | − 4| > 3 =⇒ | − 4| > 2.
Proof. Assume that | − 4| > 3. Since 3 > 2, we have | − 4| > 3 > 2, so
| − 4| > 2.
2.1.5 ̸
p =⇒ q
If we want to show p does not imply q, we must show that p is true and q is false
at the same time, which means the question breaks into two questions: one is
to show p is true, the other is to show q is false.
̸
Exercise 5. Show: 2 > 1 =⇒ 1 = 0.
Proof. It suffices to show 2 > 1 and 1 ̸= 0.
1. First, 2 > 1.
2. Also, 1 ̸= 0.
̸
This tells us that 2 > 1 =⇒ 1 = 0.
2
Maki Lu
October 11, 2023
1 Introduction
1.1 Purpose of the Review Notes
I am Maki, a 2nd year math PhD student and a TA in this course for TUT0402
and TUT0803. Some of my students from my tutorials prepared some review
notes and want to share with everyone in the course, so I am glad to type it
down using LaTeX and add some important tips so that you can take advantage
of this file for the term test.
1.2 Reference
The reference is my students’ summaries of what they have learned from the
lectures, tutorials, textbooks, videos, practice problems and some past problem
sets. This may not be a full list, but it should be helpful in some sense.
I am proud of my students, and I am proud to share these notes to all of
you, and I want these notes to be helpful. If you find any typos, you can email
me.
2 How to write proofs?
In each subsection, we give the instructions to prove statements with examples.
2.1 Simple Statements
2.1.1 p
To show p directly, we might do some calculations, and in the end deduce p.
Exercise 1. Show: 32 + 3 > 10.
Proof. Note that 32 + 3 = 12 > 10, so the statement is true.
1
, 2.1.2 p and q
If we want to show p and q, it breaks into two questions: one is to show p is
true, and the other is to show q is true.
Exercise 2. Show: 32 + 3 > 10 and 3 > 1.
Proof.
1. First, 32 + 3 = 12 > 10.
2. Also, 3 is clearly bigger than 1.
This tells us that the statement is true.
2.1.3 p or q
If we want to show p or q, it suffices to show that at least one of them is true.
Exercise 3. Show: 1 > 3 or 3 > 1.
Proof. Clearly, 3 > 1. The second statement is true. So the entire statement is
true.
2.1.4 p =⇒ q
If we want to show p implies q, it suffices to first assume that p is true, and use
the assumption of p to prove that q is true.
Exercise 4. Show: | − 4| > 3 =⇒ | − 4| > 2.
Proof. Assume that | − 4| > 3. Since 3 > 2, we have | − 4| > 3 > 2, so
| − 4| > 2.
2.1.5 ̸
p =⇒ q
If we want to show p does not imply q, we must show that p is true and q is false
at the same time, which means the question breaks into two questions: one is
to show p is true, the other is to show q is false.
̸
Exercise 5. Show: 2 > 1 =⇒ 1 = 0.
Proof. It suffices to show 2 > 1 and 1 ̸= 0.
1. First, 2 > 1.
2. Also, 1 ̸= 0.
̸
This tells us that 2 > 1 =⇒ 1 = 0.
2
