AMP STUDY NOTES
MAT2613 · REAL ANALYSIS
Complete Module
Additional Practice Problems
All seven chapters’ practice problems 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
PRACTICE PROBLEMS · 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
Additional Practice Problems
Ten practice problems in the style commonly seen in MAT2613
assessments, spanning proof methods, negation, set identities, and
injective/surjective/bijective functions — each with a complete,
independently worked solution.
PRACTICE PROBLEMS · CHAPTER 1
Independent study material. These are original practice problems, written to match the style and difficulty of
typical MAT2613 assessments. They are not reproductions of any specific past exam, assignment, or memo, are
not affiliated with, endorsed by, or sourced from UNISA or any official assessment body, and include
independently worked, verified solutions. © AMP Study Notes.
, AMP Study Notes — MAT2613 — Chapter 1 Practice Problems
Chapter 1: Additional Practice Problems
Ten problems covering the full spread of Chapter 1's exam-relevant skills, each with a complete solution. Attempt
each problem before reading its solution.
Problem 1 — Direct Proof
Use a direct proof to show that the sum of any two even integers is even.
Solution. Let and be even integers, so and for some integers . Then:
Since is an integer, has the form .
Conclusion: is even.
Problem 2 — Contrapositive Proof
Prove: if is odd, then is odd. (Use the contrapositive.)
Solution. The contrapositive of " odd odd" is " even even." Assume is even, so
for some integer . Then:
Since is an integer, is even. This proves the contrapositive, which is logically equivalent to the
original statement.
Conclusion: if is odd, then is odd.
Problem 3 — Proof by Contradiction (Irrationality)
Prove that is irrational.
Solution. Assume, for contradiction, that is rational: with integers, ,
. Squaring:
So , and since is prime, . Write :
So , hence also. But then divides both and , contradicting .
Conclusion: is irrational.
Page 2 of 5
, AMP Study Notes — MAT2613 — Chapter 1 Practice Problems
Problem 4 — Proof by Contradiction (No Integer Solutions)
Use proof by contradiction to show that there do not exist integers such that .
Solution. Assume, for contradiction, that integers exist with . Since and ,
the left side satisfies:
so , meaning . But is false (no integer multiple of equals ), which is a
contradiction.
Conclusion: no integers satisfy .
Problem 5 — Negation of a Universally Quantified Statement
Write the negation of: " for all ."
Solution. This has the form " for all ," which negates to "there exists such that not
."
Answer: "There exists such that ."
Problem 6 — Negation of an Implication
Write the negation of: "If is a real number, then ."
Solution. The negation of " " is " and (not )" — an implication is only false when its antecedent
is true and its consequence is false. Here is " is a real number" and is " ," so not is
" ."
Answer: " is a real number and ."
Page 3 of 5
MAT2613 · REAL ANALYSIS
Complete Module
Additional Practice Problems
All seven chapters’ practice problems 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
PRACTICE PROBLEMS · 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
Additional Practice Problems
Ten practice problems in the style commonly seen in MAT2613
assessments, spanning proof methods, negation, set identities, and
injective/surjective/bijective functions — each with a complete,
independently worked solution.
PRACTICE PROBLEMS · CHAPTER 1
Independent study material. These are original practice problems, written to match the style and difficulty of
typical MAT2613 assessments. They are not reproductions of any specific past exam, assignment, or memo, are
not affiliated with, endorsed by, or sourced from UNISA or any official assessment body, and include
independently worked, verified solutions. © AMP Study Notes.
, AMP Study Notes — MAT2613 — Chapter 1 Practice Problems
Chapter 1: Additional Practice Problems
Ten problems covering the full spread of Chapter 1's exam-relevant skills, each with a complete solution. Attempt
each problem before reading its solution.
Problem 1 — Direct Proof
Use a direct proof to show that the sum of any two even integers is even.
Solution. Let and be even integers, so and for some integers . Then:
Since is an integer, has the form .
Conclusion: is even.
Problem 2 — Contrapositive Proof
Prove: if is odd, then is odd. (Use the contrapositive.)
Solution. The contrapositive of " odd odd" is " even even." Assume is even, so
for some integer . Then:
Since is an integer, is even. This proves the contrapositive, which is logically equivalent to the
original statement.
Conclusion: if is odd, then is odd.
Problem 3 — Proof by Contradiction (Irrationality)
Prove that is irrational.
Solution. Assume, for contradiction, that is rational: with integers, ,
. Squaring:
So , and since is prime, . Write :
So , hence also. But then divides both and , contradicting .
Conclusion: is irrational.
Page 2 of 5
, AMP Study Notes — MAT2613 — Chapter 1 Practice Problems
Problem 4 — Proof by Contradiction (No Integer Solutions)
Use proof by contradiction to show that there do not exist integers such that .
Solution. Assume, for contradiction, that integers exist with . Since and ,
the left side satisfies:
so , meaning . But is false (no integer multiple of equals ), which is a
contradiction.
Conclusion: no integers satisfy .
Problem 5 — Negation of a Universally Quantified Statement
Write the negation of: " for all ."
Solution. This has the form " for all ," which negates to "there exists such that not
."
Answer: "There exists such that ."
Problem 6 — Negation of an Implication
Write the negation of: "If is a real number, then ."
Solution. The negation of " " is " and (not )" — an implication is only false when its antecedent
is true and its consequence is false. Here is " is a real number" and is " ," so not is
" ."
Answer: " is a real number and ."
Page 3 of 5
