TMUA Learning·Mathematical Reasoning
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Develop the logical and proof-based reasoning needed to construct, test and communicate mathematical arguments.

Chapter 1. Implication and Direction

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Topic 1. Reading and Reversing Implications
10 Practices
PUBLIC
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$f$
Topic 2. Comparing Mathematical Statements
11 Practices
MEMBERLogin required
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$x$
Topic 3. Connecting Implications to Reach a Conclusion
9 Practices
MEMBERLogin required
Exercise
20 Questions
Published in book

Chapter 2. Counterexamples and Disproof

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$Topic 1 — Recognising a Valid Counterexample In Chapter 1, you learned to keep track of the direc...$
Topic 1. Recognising a Valid Counterexample
8 Practices
Published in book
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$Topic 2 — Identifying the Allowed Cases A value can satisfy the condition of a claim and make its...$
Topic 2. Identifying the Allowed Cases
10 Practices
Published in book
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$Topic 3 — Where to Search for Counterexamples Knowing what a counterexample must do is only the f...$
Topic 3. Where to Search for Counterexamples
10 Practices
Published in book
Exercise
20 Questions
Published in book

Chapter 3. Necessary and Sufficient Conditions

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$Topic 1 — Necessary and Sufficient Conditions A condition can be necessary, sufficient, b...$
Topic 1. Necessary and Sufficient Conditions
10 Practices
Published in book
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$Topic 2 — Testing Necessary and Sufficient Conditions When the relationship between two condition...$
Topic 2. Testing Necessary and Sufficient Conditions
14 Practices
Published in book
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$Topic 3 — Combining Conditions A conclusion may depend on more than one condition. When several...$
Topic 3. Combining Conditions
10 Practices
Published in book
Exercise
20 Questions
Published in book

Chapter 4. Quantifiers and Negation

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$Topic 1 — Reading Quantified Statements Mathematical statements often use words such as every ...$
Topic 1. Reading Quantified Statements
10 Practices
Published in book
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$Topic 2 — Negating Quantified Statements To negate a mathematical statement, find a statement tha...$
Topic 2. Negating Quantified Statements
10 Practices
Published in book
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$Topic 3 — The Order of Quantifiers When a statement contains more than one quantifier, the order...$
Topic 3. The Order of Quantifiers
10 Practices
Published in book
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$Topic 4 — Proving and Disproving Quantified Claims Different quantified claims require different...$
Topic 4. Proving and Disproving Quantified Claims
10 Practices
Published in book
Exercise
20 Questions
Published in book

Chapter 5. Checking Mathematical Arguments

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$Topic 1 — Checking Whether Each Step Is Justified A mathematical argument is a sequence of steps....$
Topic 1. Checking Whether Each Step Is Justified
10 Practices
Published in book
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$Topic 2 — Checking Whether Rewriting Keeps the Same Solutions When you replace an equation or ine...$
Topic 2. Checking Whether Rewriting Keeps the Same Solutions
11 Practices
Published in book
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$Topic 3 — Checking What the Argument Actually Proves An argument can contain correct mathematical...$
Topic 3. Checking What the Argument Actually Proves
10 Practices
Published in book
Exercise
20 Questions
Published in book

Chapter 6. Reasoning by Cases

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$Topic 1 — Finding All the Relevant Cases Some TMUA problems do not tell you which cases to consid...$
Topic 1. Finding All the Relevant Cases
8 Practices
Published in book
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$Topic 2 — Ruling Out Impossible Cases Finding all the branches is only the first part of reasonin...$
Topic 2. Ruling Out Impossible Cases
10 Practices
Published in book
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$Topic 3 — Following Every Remaining Case to a Conclusion After impossible cases have been removed...$
Topic 3. Following Every Remaining Case to a Conclusion
10 Practices
Published in book
Exercise
20 Questions
Published in book

TMUA Paper 2 Mock Papers

Mock 1
20 Questions
Published in book
Mock 2
20 Questions
Published in book

Chapter 1. Implication and Direction

iTopic 1. Reading and Reversing Implications

iTopic 2 — Comparing Mathematical Statements Consider two statements about a function fff: | Stat...Topic 2. Comparing Mathematical Statements

Topic 2 — Comparing Mathematical Statements

iTopic 3 — Connecting Implications to Reach a Conclusion Suppose xxx is real and you know: If... Topic 3. Connecting Implications to Reach a Conclusion

Topic 3 — Connecting Implications to Reach a Conclusion

Exercise

Chapter 2. Counterexamples and Disproof

iTopic 1 — Recognising a Valid Counterexample In Chapter 1, you learned to keep track of the direc...Topic 1. Recognising a Valid Counterexample

Topic 1 — Recognising a Valid Counterexample

iTopic 2 — Identifying the Allowed Cases A value can satisfy the condition of a claim and make its...Topic 2. Identifying the Allowed Cases

Topic 2 — Identifying the Allowed Cases

iTopic 3 — Where to Search for Counterexamples Knowing what a counterexample must do is only the f...Topic 3. Where to Search for Counterexamples

Topic 3 — Where to Search for Counterexamples

Exercise

Chapter 3. Necessary and Sufficient Conditions

iTopic 1 — Necessary and Sufficient Conditions A condition can be necessary, sufficient, b...Topic 1. Necessary and Sufficient Conditions

Topic 1 — Necessary and Sufficient Conditions

iTopic 2 — Testing Necessary and Sufficient Conditions When the relationship between two condition...Topic 2. Testing Necessary and Sufficient Conditions

Topic 2 — Testing Necessary and Sufficient Conditions

iTopic 3 — Combining Conditions A conclusion may depend on more than one condition. When several...Topic 3. Combining Conditions

Topic 3 — Combining Conditions

Exercise

Chapter 4. Quantifiers and Negation

iTopic 1 — Reading Quantified Statements Mathematical statements often use words such as every...Topic 1. Reading Quantified Statements

Topic 1 — Reading Quantified Statements

iTopic 2 — Negating Quantified Statements To negate a mathematical statement, find a statement tha...Topic 2. Negating Quantified Statements

Topic 2 — Negating Quantified Statements

iTopic 3 — The Order of Quantifiers When a statement contains more than one quantifier, the order...Topic 3. The Order of Quantifiers

Topic 3 — The Order of Quantifiers

iTopic 4 — Proving and Disproving Quantified Claims Different quantified claims require different...Topic 4. Proving and Disproving Quantified Claims

Topic 4 — Proving and Disproving Quantified Claims

Exercise

Chapter 5. Checking Mathematical Arguments

iTopic 1 — Checking Whether Each Step Is Justified A mathematical argument is a sequence of steps....Topic 1. Checking Whether Each Step Is Justified

Topic 1 — Checking Whether Each Step Is Justified

iTopic 2 — Checking Whether Rewriting Keeps the Same Solutions When you replace an equation or ine...Topic 2. Checking Whether Rewriting Keeps the Same Solutions

Topic 2 — Checking Whether Rewriting Keeps the Same Solutions

iTopic 3 — Checking What the Argument Actually Proves An argument can contain correct mathematical...Topic 3. Checking What the Argument Actually Proves

Topic 3 — Checking What the Argument Actually Proves

Exercise

Chapter 6. Reasoning by Cases

iTopic 1 — Finding All the Relevant Cases Some TMUA problems do not tell you which cases to consid...Topic 1. Finding All the Relevant Cases

Topic 1 — Finding All the Relevant Cases

iTopic 2 — Ruling Out Impossible Cases Finding all the branches is only the first part of reasonin...Topic 2. Ruling Out Impossible Cases

Topic 2 — Ruling Out Impossible Cases

iTopic 3 — Following Every Remaining Case to a Conclusion After impossible cases have been removed...Topic 3. Following Every Remaining Case to a Conclusion

Topic 3 — Following Every Remaining Case to a Conclusion

Exercise

TMUA Paper 2 Mock Papers

Mock 1

Mock 2

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Topic 1. Reading and Reversing Implications

i
$f$
Topic 2. Comparing Mathematical Statements

i
$x$
Topic 3. Connecting Implications to Reach a Conclusion

i
$Topic 1 — Recognising a Valid Counterexample In Chapter 1, you learned to keep track of the direc...$
Topic 1. Recognising a Valid Counterexample

i
$Topic 2 — Identifying the Allowed Cases A value can satisfy the condition of a claim and make its...$
Topic 2. Identifying the Allowed Cases

i
$Topic 3 — Where to Search for Counterexamples Knowing what a counterexample must do is only the f...$
Topic 3. Where to Search for Counterexamples

i
$Topic 1 — Necessary and Sufficient Conditions A condition can be necessary, sufficient, b...$
Topic 1. Necessary and Sufficient Conditions

i
$Topic 2 — Testing Necessary and Sufficient Conditions When the relationship between two condition...$
Topic 2. Testing Necessary and Sufficient Conditions

i
$Topic 3 — Combining Conditions A conclusion may depend on more than one condition. When several...$
Topic 3. Combining Conditions

i
$Topic 1 — Reading Quantified Statements Mathematical statements often use words such as every ...$
Topic 1. Reading Quantified Statements

i
$Topic 2 — Negating Quantified Statements To negate a mathematical statement, find a statement tha...$
Topic 2. Negating Quantified Statements

i
$Topic 3 — The Order of Quantifiers When a statement contains more than one quantifier, the order...$
Topic 3. The Order of Quantifiers

i
$Topic 4 — Proving and Disproving Quantified Claims Different quantified claims require different...$
Topic 4. Proving and Disproving Quantified Claims

i
$Topic 1 — Checking Whether Each Step Is Justified A mathematical argument is a sequence of steps....$
Topic 1. Checking Whether Each Step Is Justified

i
$Topic 2 — Checking Whether Rewriting Keeps the Same Solutions When you replace an equation or ine...$
Topic 2. Checking Whether Rewriting Keeps the Same Solutions

i
$Topic 3 — Checking What the Argument Actually Proves An argument can contain correct mathematical...$
Topic 3. Checking What the Argument Actually Proves

i
$Topic 1 — Finding All the Relevant Cases Some TMUA problems do not tell you which cases to consid...$
Topic 1. Finding All the Relevant Cases

i
$Topic 2 — Ruling Out Impossible Cases Finding all the branches is only the first part of reasonin...$
Topic 2. Ruling Out Impossible Cases

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$Topic 3 — Following Every Remaining Case to a Conclusion After impossible cases have been removed...$
Topic 3. Following Every Remaining Case to a Conclusion