Chapter 1 — Implication and Direction

Mathematical statements often tell you that one condition leads to another.

For example:

If $x>8$, then $x>4$.

In TMUA questions, this relationship may be written using if, only if or if and only if. You may need to check whether the reverse is true, compare what two statements tell you, or connect several statements to reach a conclusion.

What am I given, and what does it allow me to conclude?

Keeping the direction clear prevents you from using a statement backwards or joining statements that do not form a valid route.

In this chapter, you will learn to:

• read if, only if and if and only if in the correct direction;
• identify a converse and use a contrapositive when it gives the direction you need;
• compare statements by checking what each one guarantees;
• connect implications to reach a required conclusion;
• use mathematical deductions to bridge gaps between implications.

Chapter 2 — Counterexamples and Disproof

A mathematical statement may look convincing and still be false.

For example:

If a triangle has two equal sides, then it has a right angle.

Consider a triangle with angles $50^\circ,50^\circ,80^\circ$. The two equal angles mean that the opposite sides are equal, so the condition is satisfied. But the triangle has no right angle.

Therefore the statement is false.

An example that disproves a claim is called a counterexample. For an if–then statement, a counterexample must satisfy the condition and make the conclusion false.

In this chapter, you will learn to:

• recognise a valid counterexample;
• check which values or objects are allowed by the statement;
• find where the conclusion is false and use this to narrow the search;
• check boundaries, opposite signs, zero cases and other useful cases;
• determine when no counterexample exists.

To disprove “If $P$, then $Q$”, look for an allowed case where $P$ is true and $Q$ is false.

Chapter 3 — Necessary and Sufficient Conditions

Suppose we want to know whether an integer is divisible by $6$.

Being divisible by $3$ is necessary: every multiple of $6$ must be divisible by $3$. But it is not sufficient, since $9$ is divisible by $3$ but not by $6$.

Being divisible by $12$ is sufficient: every multiple of $12$ is divisible by $6$. But it is not necessary, since $6$ itself is not divisible by $12$.

So necessary and sufficient ask different questions:

• Necessary: Must this condition be true for the conclusion to be true?
• Sufficient: Does this condition guarantee the conclusion?

A condition can be necessary, sufficient, both or neither. Deciding the relationship often requires checking two directions separately.

In this chapter, you will learn to:

• test the two directions separately;
• prove a direction or disprove it with a counterexample;
• determine whether a condition is necessary, sufficient, both or neither;
• apply several conditions together and determine what they guarantee.

Check both directions separately. One direction does not automatically give you the other.

Chapter 4 — Quantifiers and Negation

Mathematical statements often make claims about every case, at least one case, or no case.

For example:

Every root of the equation is positive.

and

Some root of the equation is positive.

make different claims. The first must hold for all the roots, while the second requires at least one positive root.

In TMUA questions, quantified information may be combined with inequalities, functions, equations or restrictions on the allowed values. You may need to determine what must follow, identify exactly what would make a statement false, decide whether one choice must work in every case, or establish whether a required example exists.

A useful question throughout this chapter is:

Which cases does the statement cover, and what does it require of them?

Answering this question helps determine what kind of reasoning the statement requires.

In this chapter, you will learn to:

• read every, some, at least one and none with the correct domain and restrictions;
• determine what must, may or need not follow from quantified information;
• negate quantified statements completely, including inequalities and and/or conditions;
• distinguish a choice that may depend on an earlier value from one fixed choice that must work for every case;
• prove or disprove quantified claims using suitable examples, counterexamples or arguments covering all allowed cases;
• use an opposite quantified claim when it makes the mathematics easier.

Chapter 5 — Checking Mathematical Arguments

A mathematical argument can look convincing even when one step is not justified.

For example, suppose a student writes $\frac{x+2}{x}>3$ and immediately multiplies by $x$. The algebra may look routine, but the direction of the inequality depends on whether $x$ is positive or negative. A hidden condition can change whether the whole argument works.

Similar problems arise in many forms. Cancelling a factor may lose a possible case. Squaring may introduce values that were not solutions of the original equation. Taking a square root may require an absolute value. An argument may prove a bound without showing that the bound is attained, or reach a true conclusion by an invalid route.

In TMUA questions, you therefore need to check more than whether the final answer looks reasonable.

A useful question throughout this chapter is:

What exactly justifies this step, and does the information given guarantee that it is valid?

Sometimes earlier information determines whether an operation is valid by fixing the sign, domain or possible exceptional values. Sometimes you need to decide whether rewriting an equation or inequality preserves the same solutions. In other questions, every calculation may be correct, but the argument may still fail to justify the final conclusion.

In this chapter, you will learn to:

• check whether each step in an argument is justified and identify the first step that fails;
• use earlier conditions to check signs, domains and exceptional values;
• recognise when operations such as cancelling, squaring, taking roots or using logarithms change or preserve the possible solutions;
• distinguish between what an argument has proved and what its final conclusion claims;
• recognise when a correct conclusion has been reached by an invalid argument.

Chapter 6 — Reasoning by Cases

Some mathematical problems cannot be completed by following one line of reasoning from start to finish.

The information may allow several different values, signs, orders or geometric configurations. One case may later turn out to be impossible. Another may split again. Two different cases may even lead to the same final answer.

For example, suppose $x^2=9$.

Then $x=3 \quad\text{or}\quad x=-3$.

If a later condition involves $x$, you cannot simply choose the value that looks more convenient. Both possibilities must be considered unless another condition rules one of them out.

The same issue can be less obvious. A trigonometric equation may allow an acute and an obtuse angle. Two roots may be assigned to variables in different orders. A diagram may show only one of several possible configurations. Several independent choices may combine to create more cases than expected.

In TMUA questions, the difficulty is often not the calculation inside one case. It is deciding:

• what the possible cases are;
• whether every possibility has been included;
• which cases are compatible with all the conditions;
• what each surviving case leads to;
• whether different cases give different or identical final outcomes.

A useful question throughout this chapter is:

Have I accounted for every possible case allowed by the information?

Reasoning by cases is different from finding a counterexample. To disprove a statement, one valid counterexample may be enough. In a case problem, finding one possible case is not enough if other cases remain.

It is also different from checking whether the steps in an argument are justified. Every step may be valid while the overall solution is still incomplete because one possible branch was ignored.

In this chapter, you will learn to:

• identify all the cases allowed by the information;
• combine separate choices without losing possibilities;
• use later conditions to rule out impossible cases;
• follow every surviving case, including cases that split again;
• use symmetry while keeping genuinely different outcomes separate;
• combine the final outcomes correctly.

The chapter follows three stages:

1. Finding All the Relevant Cases — identify every possible case.
2. Ruling Out Impossible Cases — remove cases that cannot satisfy all the conditions.
3. Following Every Remaining Case to a Conclusion — follow the surviving cases and combine their outcomes.

A complete case solution deals with every relevant case.

Instructions

Time allowed: 75 minutes
Questions: 20 multiple-choice questions

• Do not use a calculator or dictionary.
• Choose one answer for each question.
• There is no penalty for an incorrect answer, so attempt every question.
• Complete the paper under timed conditions before checking the Answer Key or Full Solutions.