AMP STUDY NOTES
MAT2613 · REAL ANALYSIS
Complete Module
Study Notes
All seven chapters’ study notes combined into a single volume, in
order, for the full module.
Chapter 1 — Preliminaries
Chapter 2 — The Real Numbers
Chapter 3 — Sequences
Chapter 4 — Series
Chapter 5 — Continuous Functions
Chapter 6 — Differentiation
Chapter 7 — Integration
STUDY NOTES · ALL CHAPTERS
Independent study material. Independently authored revision aid, not affiliated with, endorsed by, or sourced
from UNISA or any official assessment body. © AMP Study Notes.
,AMP STUDY NOTES
MAT2613 · REAL ANALYSIS
Chapter 1
Preliminaries
Exam-focused study notes: logical statements and implications, the
three core proof methods, set operations, and injective/surjective/
bijective functions — with fully worked practice problems and
complete step-by-step solutions.
STUDY NOTES · CHAPTER 1
Independent study material. These notes are an independently authored revision aid based on the standard
undergraduate real analysis syllabus (following the treatment in Haggarty, Fundamentals of Mathematical
Analysis) and on general patterns observed in how this material tends to be assessed. They are not affiliated with,
endorsed by, or sourced from the University of South Africa (UNISA) or any official assessment body, and all
practice problems are original variations written for study purposes, with independently worked solutions. © AMP
Study Notes.
, AMP Study Notes — MAT2613 — Chapter 1
Chapter 1: Preliminaries
TOPICS COVERED IN THIS CHAPTER
• Implications, converse, and contrapositive statements • Sets and set operations
• Negating statements (including quantified statements) • Functions: domain, codomain, image
• The three core methods of proof: direct, • Injective, surjective, and bijective functions, and
contrapositive, and contradiction inverses
MAT2613 Real Analysis (Unisa). This opening chapter is the toolkit every later chapter leans on —
almost every proof from Chapter 3 onward uses one of the three proof methods introduced here, and
"first principles" limit/continuity proofs later in the module are themselves just contrapositive- or
direct-style arguments dressed in - notation. A review of recent exam sittings shows this chapter is
tested almost every sitting through a mix of a direct/contradiction/contrapositive proof question, a
"negate this statement" question, and a question asking whether a given function is injective/surjective/
bijective (often paired with finding its inverse).
What is Most Examined?
HIGH-YIELD TASK HOW IT TENDS TO APPEAR PRIORITY
Negating a statement Write the negation of a given implication, universally/existentially Very high
quantified statement, or compound statement.
Proof by contradiction Prove an irrationality result or a "no integer solutions exist" result by Very high
assuming the opposite and deriving a contradiction.
Bijective functions and Show a given (often piecewise-patched) function is bijective, then find High
inverses its inverse explicitly.
Direct and Prove a simple number-theoretic or set-theoretic statement using direct High
contrapositive proofs proof, or its logically equivalent contrapositive.
Image of a set under a Find for a given interval and (often piecewise or absolute- Medium-
function value) function . high
Table of Contents
1. Implications, Converse, and Contrapositive
2. Negating Statements
3. Methods of Proof
Page 2 of 8
, AMP Study Notes — MAT2613 — Chapter 1
4. Sets and Set Operations
5. Functions: Domain, Codomain, and Image
6. Injective, Surjective, and Bijective Functions
7. Worked Examples
8. Common Mistakes
9. Exam Checklist
10. Key Results Summary
Implications, Converse, and Contrapositive
An implication " " (read " implies ," or "if then ") asserts that whenever the statement (the
antecedent) is true, the statement (the consequence) must also be true. Analysis is built almost entirely out of
chains of implications, so being fluent in how they relate to each other is foundational.
CONVERSE AND CONTRAPOSITIVE
• The converse of is — swapping antecedent and consequence. The converse of a true
implication is not automatically true.
• The contrapositive of is . Unlike the converse, the contrapositive is
always logically equivalent to the original implication — proving one proves the other.
COMMON MISTAKE
Confusing the converse with the contrapositive is one of the most common errors in this chapter. A classic
illustration: "if is differentiable at then is continuous at " is true, and its contrapositive ("if is not
continuous at then is not differentiable at ") is therefore also true — but its converse ("if is
continuous at then is differentiable at ") is false, since is continuous but not differentiable
at .
Negating Statements
Being able to write the precise negation of a statement is a prerequisite for both contradiction proofs (which begin
by assuming the negation) and contrapositive proofs (which restate the implication using negations). The rules are
mechanical but easy to get wrong under exam pressure.
STATEMENT NEGATION
is true is false
and (not ) or (not )
or (not ) and (not )
is true for all There exists such that is false
Page 3 of 8
MAT2613 · REAL ANALYSIS
Complete Module
Study Notes
All seven chapters’ study notes combined into a single volume, in
order, for the full module.
Chapter 1 — Preliminaries
Chapter 2 — The Real Numbers
Chapter 3 — Sequences
Chapter 4 — Series
Chapter 5 — Continuous Functions
Chapter 6 — Differentiation
Chapter 7 — Integration
STUDY NOTES · ALL CHAPTERS
Independent study material. Independently authored revision aid, not affiliated with, endorsed by, or sourced
from UNISA or any official assessment body. © AMP Study Notes.
,AMP STUDY NOTES
MAT2613 · REAL ANALYSIS
Chapter 1
Preliminaries
Exam-focused study notes: logical statements and implications, the
three core proof methods, set operations, and injective/surjective/
bijective functions — with fully worked practice problems and
complete step-by-step solutions.
STUDY NOTES · CHAPTER 1
Independent study material. These notes are an independently authored revision aid based on the standard
undergraduate real analysis syllabus (following the treatment in Haggarty, Fundamentals of Mathematical
Analysis) and on general patterns observed in how this material tends to be assessed. They are not affiliated with,
endorsed by, or sourced from the University of South Africa (UNISA) or any official assessment body, and all
practice problems are original variations written for study purposes, with independently worked solutions. © AMP
Study Notes.
, AMP Study Notes — MAT2613 — Chapter 1
Chapter 1: Preliminaries
TOPICS COVERED IN THIS CHAPTER
• Implications, converse, and contrapositive statements • Sets and set operations
• Negating statements (including quantified statements) • Functions: domain, codomain, image
• The three core methods of proof: direct, • Injective, surjective, and bijective functions, and
contrapositive, and contradiction inverses
MAT2613 Real Analysis (Unisa). This opening chapter is the toolkit every later chapter leans on —
almost every proof from Chapter 3 onward uses one of the three proof methods introduced here, and
"first principles" limit/continuity proofs later in the module are themselves just contrapositive- or
direct-style arguments dressed in - notation. A review of recent exam sittings shows this chapter is
tested almost every sitting through a mix of a direct/contradiction/contrapositive proof question, a
"negate this statement" question, and a question asking whether a given function is injective/surjective/
bijective (often paired with finding its inverse).
What is Most Examined?
HIGH-YIELD TASK HOW IT TENDS TO APPEAR PRIORITY
Negating a statement Write the negation of a given implication, universally/existentially Very high
quantified statement, or compound statement.
Proof by contradiction Prove an irrationality result or a "no integer solutions exist" result by Very high
assuming the opposite and deriving a contradiction.
Bijective functions and Show a given (often piecewise-patched) function is bijective, then find High
inverses its inverse explicitly.
Direct and Prove a simple number-theoretic or set-theoretic statement using direct High
contrapositive proofs proof, or its logically equivalent contrapositive.
Image of a set under a Find for a given interval and (often piecewise or absolute- Medium-
function value) function . high
Table of Contents
1. Implications, Converse, and Contrapositive
2. Negating Statements
3. Methods of Proof
Page 2 of 8
, AMP Study Notes — MAT2613 — Chapter 1
4. Sets and Set Operations
5. Functions: Domain, Codomain, and Image
6. Injective, Surjective, and Bijective Functions
7. Worked Examples
8. Common Mistakes
9. Exam Checklist
10. Key Results Summary
Implications, Converse, and Contrapositive
An implication " " (read " implies ," or "if then ") asserts that whenever the statement (the
antecedent) is true, the statement (the consequence) must also be true. Analysis is built almost entirely out of
chains of implications, so being fluent in how they relate to each other is foundational.
CONVERSE AND CONTRAPOSITIVE
• The converse of is — swapping antecedent and consequence. The converse of a true
implication is not automatically true.
• The contrapositive of is . Unlike the converse, the contrapositive is
always logically equivalent to the original implication — proving one proves the other.
COMMON MISTAKE
Confusing the converse with the contrapositive is one of the most common errors in this chapter. A classic
illustration: "if is differentiable at then is continuous at " is true, and its contrapositive ("if is not
continuous at then is not differentiable at ") is therefore also true — but its converse ("if is
continuous at then is differentiable at ") is false, since is continuous but not differentiable
at .
Negating Statements
Being able to write the precise negation of a statement is a prerequisite for both contradiction proofs (which begin
by assuming the negation) and contrapositive proofs (which restate the implication using negations). The rules are
mechanical but easy to get wrong under exam pressure.
STATEMENT NEGATION
is true is false
and (not ) or (not )
or (not ) and (not )
is true for all There exists such that is false
Page 3 of 8
