Test of Mathematics for University Admission (TMUA) Mathematical Reasoning and Proof Syllabus

Outline a direct proof that the sum of two even integers is even.

Let the integers be $2a$ and $2b$ for integers $a,b$. Their sum is $2a+2b = 2(a+b)$, which is $2 \times$ an integer, hence even. $\blacksquare$

How is an even integer and an odd integer represented algebraically in proofs?

Even: $2k$ for some integer $k$. Odd: $2k+1$ (or $2k-1$) for some integer $k$. These general forms are the starting point of most direct parity proofs.

What is proof by exhaustion?

A method that proves a statement by splitting the situation into a finite number of cases and verifying the statement holds in every case. It only works when the cases are finite and complete.

When is proof by exhaustion a valid and appropriate method?

When the domain can be partitioned into finitely many cases that together cover all possibilities, and each case can be checked. It is unsuitable for infinitely many cases.

Give an example structure of a proof by exhaustion for a statement about $n \bmod 3$.

Every integer $n$ is of the form $3k$, $3k+1$, or $3k+2$. Prove the claim separately for each of these three residue cases; since they cover all integers, the claim holds for all $n$.

What is the logical basis of proof by contradiction?

To prove $P$, assume its negation $\neg P$ is true and derive a logical contradiction (something that is both true and false). Since $\neg P$ leads to absurdity, $P$ must be true. It uses the law $\neg P \Rightarrow \text{False}$, so $P$.

To prove an implication $P \Rightarrow Q$ by contradiction, what do you assume?

Assume $P$ is true and $Q$ is false simultaneously (i.e. assume $P \land \neg Q$), then derive a contradiction.

Outline the proof by contradiction that $\sqrt{2}$ is irrational.

Assume $\sqrt{2} = \frac{a}{b}$ in lowest terms. Then $2b^{2} = a^{2}$, so $a^{2}$ is even, hence $a$ is even, say $a=2c$. Then $b^{2}=2c^{2}$, so $b$ is even too — contradicting $\frac{a}{b}$ being in lowest terms. So $\sqrt{2}$ is irrational. $\blacksquare$

How does proof by contradiction relate to proof by contrapositive?

Proving the contrapositive $\neg Q \Rightarrow \neg P$ is a special, cleaner case: you assume $\neg Q$ and derive $\neg P$. General contradiction assumes $P \land \neg Q$ and derives any absurdity, not necessarily $\neg P$.

What is disproof by counterexample, and what type of statement does it disprove?

To disprove a universal statement $\forall x \, P(x)$, you exhibit a single specific $x$ for which $P(x)$ is false. One counterexample is enough to disprove a "for all" claim.

Can a counterexample ever prove a universal statement true? Why or why not?

No. Verifying the statement for one or many examples never proves $\forall x \, P(x)$; examples can only disprove it (via a counterexample) or fail to disprove it. Proof of a universal claim requires a general argument.

Give a counterexample to the claim "every prime number is odd."

$2$ is prime and even, so it is a counterexample. The claim is false.

Give a counterexample to "if $n^{2}$ is divisible by $4$ then $n$ is divisible by $4$."

$n=2$: $n^{2}=4$ is divisible by $4$, but $n=2$ is not divisible by $4$. So the statement is false.

What does it mean to identify flawed reasoning in a proof?

Locating the specific step where a deduction does not actually follow from what precedes it — e.g. an unjustified assumption, a division by a possibly-zero quantity, affirming the converse, or a step that doesn't preserve equivalence.

Name the fallacy: from $P \Rightarrow Q$ and $Q$, concluding $P$.

Affirming the consequent. It is invalid because $Q$ can hold for reasons other than $P$; only $P \Rightarrow Q$ with $P$ (modus ponens) validly gives $Q$.

Name the fallacy: from $P \Rightarrow Q$ and $\neg P$, concluding $\neg Q$.

Denying the antecedent. Invalid: $\neg P$ tells us nothing about $Q$. The valid rule is the contrapositive (modus tollens): from $P \Rightarrow Q$ and $\neg Q$, conclude $\neg P$.

State modus ponens and modus tollens.

Modus ponens: from $P \Rightarrow Q$ and $P$, conclude $Q$. Modus tollens: from $P \Rightarrow Q$ and $\neg Q$, conclude $\neg P$. Both are valid.

Why is squaring both sides of an equation a common source of flawed reasoning?

Squaring is not a reversible (one-way) operation: $a=b \Rightarrow a^{2}=b^{2}$, but the converse fails since $a^{2}=b^{2}$ allows $a=-b$. Squaring can introduce extraneous solutions that must be checked.

What is the difference between a valid argument and a sound argument?

An argument is valid if the conclusion follows logically from the premises (the form is correct). It is sound if it is valid AND all its premises are actually true. Validity concerns form; soundness adds truth.

Can a valid argument have a false conclusion?

Yes — if one or more premises are false. Validity guarantees only that IF all premises are true THEN the conclusion is true. With a false premise, a valid argument can yield a false conclusion.

Can an argument with a true conclusion be invalid?

Yes. The conclusion may happen to be true even though it does not logically follow from the premises. A true conclusion does not certify valid reasoning.

Distinguish the validity of reasoning from the truth of the conclusion.

Validity is about whether the inference steps correctly preserve truth from premises to conclusion; truth of the conclusion is about the actual state of affairs. A conclusion can be true with invalid reasoning, or false with valid reasoning (false premises).

What is generalisation in mathematical reasoning?

Extending a result proven for specific cases to a broader class, justified by a general argument. Unjustified generalisation from a few examples is a fallacy (hasty generalisation); valid generalisation requires a proof covering all instances.

What role do special cases play in reasoning?

Special cases can suggest conjectures, test plausibility, or disprove a general claim via counterexample. But confirming special cases never proves a general statement — it can only refute it or motivate a proof.

Why must boundary or special cases (like $0$, equality, or empty set) be checked separately?

General arguments sometimes assume a quantity is nonzero, positive, or that strict inequality holds; the special/boundary case may behave differently and break the reasoning, so it must be verified on its own.

What does it mean to make deductions from given information?

Combining stated facts/premises using valid logical rules to reach conclusions that necessarily follow, without adding unstated assumptions. Each deduced fact must be entailed by the given information.

If you are told $x$ is an integer with $3 < x \le 6$, what can you validly deduce about $x$?

$x \in \{4,5,6\}$. The constraint together with integrality restricts $x$ to exactly these three values; nothing more can be deduced without further information.

For the quadratic $ax^{2}+bx+c=0$ ($a\neq 0$), what condition on the discriminant gives two distinct real roots, one repeated root, or no real roots?

With discriminant $\Delta = b^{2}-4ac$: $\Delta > 0$ gives two distinct real roots; $\Delta = 0$ gives one repeated (equal) real root; $\Delta < 0$ gives no real roots (complex roots).

What condition on $k$ ensures $x^{2}+kx+9=0$ has no real solutions?

Need discriminant $< 0$: $k^{2}-4(1)(9) < 0 \Rightarrow k^{2} < 36 \Rightarrow -6 < k < 6$.

When solving inequalities, why must you reverse the inequality sign, and when does a sign reversal occur?

Multiplying or dividing both sides by a negative quantity reverses the inequality (e.g. $-2x < 6 \Rightarrow x > -3$). If the multiplier's sign is unknown (e.g. multiplying by $x$), you must split into cases or avoid it, or the reasoning is flawed.

How can you represent the set of solutions to an inequality, and what does combining two inequalities with "and" versus "or" mean for the solution sets?

As a set or interval on the number line. "$A$ and $B$" is the intersection $A \cap B$ (values satisfying both); "$A$ or $B$" is the union $A \cup B$ (values satisfying at least one). Reasoning about solutions means manipulating these sets correctly.