Mathematics Admissions Test (MAT) Logic, Proof and Mathematical Reasoning Flashcards

24 sample cards from the Logic, Proof and Mathematical Reasoning deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

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

   Assume $\sqrt{2} = \frac{a}{b}$ in lowest terms. Then $a^{2} = 2b^{2}$, so $a$ is even, $a = 2c$, giving $b^{2} = 2c^{2}$, so $b$ is also even. Then $a,b$ share factor $2$, contradicting "lowest terms." Hence $\sqrt{2}$ is irrational.

2. Sketch Euclid's proof by contradiction that there are infinitely many primes.

   Assume finitely many primes $p_{1},\dots,p_{n}$. Form $N = p_{1}p_{2}\cdots p_{n} + 1$. $N$ leaves remainder $1$ on division by each $p_{i}$, so it has a prime factor not in the list — a contradiction. Hence infinitely many primes.

3. How does proof by contradiction differ from proof by contrapositive?

   Contrapositive proves $P \implies Q$ by directly proving $\lnot Q \implies \lnot P$. Contradiction assumes $P \land \lnot Q$ and seeks any contradiction. Contrapositive is a special, cleaner case when the contradiction is exactly $\lnot P$.

4. What is proof by exhaustion?

   A method that splits the problem into a finite number of cases and verifies the statement in each case. It works only when the cases are finite and collectively cover all possibilities.

5. Use proof by exhaustion to show $n^{2} \equiv 0$ or $1 \pmod 4$ for every integer $n$.

   Every integer is even or odd. If $n = 2k$, $n^{2} = 4k^{2} \equiv 0$. If $n = 2k+1$, $n^{2} = 4k^{2}+4k+1 \equiv 1$. The two cases exhaust all integers, so $n^{2} \equiv 0$ or $1 \pmod 4$.

6. What is the basic idea of proof by mathematical induction?

   To prove $P(n)$ for all integers $n \geq n_{0}$: prove the base case $P(n_{0})$, then prove the inductive step that $P(k) \implies P(k+1)$. Together these give $P(n)$ for all $n \geq n_{0}$.

7. What two components must every induction proof contain?

   A base case (verify the statement for the smallest value, often $n=1$) and an inductive step (assume $P(k)$, the inductive hypothesis, and prove $P(k+1)$).

8. State the formula for $\sum_{r=1}^{n} r$ that is commonly proved by induction.

   $\displaystyle \sum_{r=1}^{n} r = \frac{n(n+1)}{2}$.

9. In an induction proof, what is the "inductive hypothesis"?

   It is the assumption that the statement $P(k)$ holds for some particular integer $k$; you use it to deduce $P(k+1)$.

10. Why is the base case essential in induction (what goes wrong without it)?

    Without a verified base case the inductive step only chains assumptions together with no starting truth, so the whole argument is unsupported — you could "prove" false statements like $P(k)\implies P(k+1)$ for a statement that is never actually true.

11. What is the formula $\sum_{r=1}^{n} r^{2}$ equal to?

    $\displaystyle \sum_{r=1}^{n} r^{2} = \frac{n(n+1)(2n+1)}{6}$.

12. When given an unfamiliar definition in a MAT problem, what should you do first?

    Read it precisely and test it on small/simple concrete cases to understand exactly what it does, rather than assuming it matches a familiar concept with a similar name.

13. If a problem defines an operation $a * b = a + b - ab$, how do you evaluate $2 * 3$?

    Substitute directly into the definition: $2 * 3 = 2 + 3 - (2)(3) = 5 - 6 = -1$.

14. A problem defines $f$ to be "self-inverse" if $f(f(x)) = x$ for all $x$. How do you check whether a given $f$ satisfies this?

    Compute the composition $f(f(x))$ by substituting $f(x)$ into $f$, simplify, and verify it equals $x$ for all $x$ in the domain.

15. When working with an unfamiliar definition, why is it useful to check edge/boundary cases?

    Edge cases (e.g. $0$, $1$, negatives, equality boundaries) often reveal how the definition behaves, expose hidden constraints, and prevent false assumptions that the definition behaves "as expected."

16. In a MAT multiple-choice question, what is the elimination strategy?

    Rule out answer options that are impossible — via counterexamples, parity/sign checks, limiting/boundary behaviour, or dimensional reasoning — narrowing to the correct choice without full computation.

17. How can substituting a convenient specific value help eliminate multiple-choice options?

    If a stated identity/expression must hold for all values, plug in an easy value (e.g. $x=0$ or $x=1$); any option that gives the wrong result there can be eliminated immediately.

18. What is estimation used for in MAT-style problems?

    To approximate a quantity's size or sign quickly so you can select or eliminate answer choices without exact calculation — e.g. bounding a value between two known numbers.

19. How can checking the sign or magnitude of an expression help eliminate options?

    If reasoning shows the answer must be, say, positive and greater than $10$, any negative or small option is eliminated, often leaving a unique choice.

20. How does considering limiting behaviour (e.g. as $x \to 0$ or $x \to \infty$) help on multiple-choice questions?

    Comparing how each candidate expression behaves in a limit against the required behaviour of the true answer eliminates options that grow, decay, or sign-change incorrectly.

21. In MAT long-answer questions, why is showing clear working important?

    Marks are awarded for method and logical steps, not just the final answer; clear working lets the examiner follow and credit your reasoning even if the final answer has a slip.

22. What is a good structure for a MAT long-answer proof or solution?

    State what you are proving/finding, set up notation and assumptions, proceed in clear logical steps with justification, and finish with a concluding statement that explicitly answers the question.

23. In a multi-part long-answer MAT question, how should later parts often be approached?

    Later parts usually build on earlier ones — reuse results, definitions, or methods established earlier (the question is scaffolded), rather than starting each part from scratch.

24. What should a proof's final line do, and why does it matter for marks?

    It should clearly state the conclusion (e.g. "hence the result follows" / restate what was proved), signalling the argument is complete; a missing or unjustified conclusion can lose the final method/accuracy marks.

What this deck covers

Logic, Proof and Mathematical Reasoning flashcards FAQ