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

Logic, Proof and Mathematical Reasoning syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Logic, Proof and Mathematical Reasoning in Mathematics Admissions Test (MAT), not a summary of it.

1. 1

   Logic and Statements

   3 topics

   • Implications and Equivalence
     • If-then statements, converse and contrapositive
     • Necessary and sufficient conditions
   • Negation and Quantifiers
     • Negating statements correctly
     • Reasoning with 'for all' and 'there exists'
   • Counterexamples
     • Disproving a general claim with a single counterexample
2. 2

   Methods of Proof

   3 topics

   • Direct Proof
     • Building a chain of valid deductions
     • Algebraic identities and divisibility proofs
   • Proof by Contradiction
     • Structure of a contradiction argument
     • Classic examples such as irrationality and infinitude of primes
   • Proof by Exhaustion and Induction (informal)
     • Checking all cases when finitely many
     • Recognising an inductive pattern in a sequence of results
3. 3

   Multiple-Choice Reasoning Strategy

   3 topics

   • Interpreting Unfamiliar Definitions
     • Applying a newly defined operation or function
     • Testing small or extreme cases to narrow options
   • Elimination and Estimation
     • Ruling out options by bounds, sign or parity
     • Sanity-checking answers for plausibility
   • Long-Answer Structure
     • Using earlier parts to scaffold later parts
     • Writing clear, justified reasoning for full credit

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

22 of 49 cards from the Logic, Proof and Mathematical Reasoning deck — real questions with worked answers.

1. What does the implication $P \implies Q$ mean?

   It means "if $P$ is true, then $Q$ is true." It makes no claim when $P$ is false. $P$ is the hypothesis (sufficient condition) and $Q$ is the conclusion (necessary condition).

2. What is the converse of the statement $P \implies Q$, and is it logically equivalent to the original?

   The converse is $Q \implies P$. It is NOT logically equivalent to $P \implies Q$; one can be true while the other is false.

3. What is the contrapositive of $P \implies Q$, and how does its truth relate to the original?

   The contrapositive is $\lnot Q \implies \lnot P$. It is logically equivalent to $P \implies Q$, so proving one proves the other.

4. What does $P \iff Q$ (equivalence / "if and only if") assert?

   It asserts that both $P \implies Q$ and $Q \implies P$ hold, so $P$ and $Q$ are true under exactly the same conditions. $P$ is necessary and sufficient for $Q$.

5. To prove an "if and only if" statement $P \iff Q$, what two things must you establish?

   You must prove both directions: $P \implies Q$ (forwards) and $Q \implies P$ (backwards / the converse).

6. In the implication $P \implies Q$, which condition is "necessary" and which is "sufficient"?

   $P$ is a sufficient condition for $Q$ (having $P$ guarantees $Q$); $Q$ is a necessary condition for $P$ (you cannot have $P$ without $Q$).

7. For which single combination of truth values is the implication $P \implies Q$ false?

   It is false only when $P$ is true and $Q$ is false. In all other cases ($P$ false, or $Q$ true) it is true.

8. What is the negation of an implication $P \implies Q$?

   $\lnot(P \implies Q)$ is equivalent to $P \land \lnot Q$, i.e. "$P$ is true and $Q$ is false."

9. State De Morgan's laws for the negation of conjunctions and disjunctions.

   $\lnot(P \land Q) \equiv \lnot P \lor \lnot Q$ and $\lnot(P \lor Q) \equiv \lnot P \land \lnot Q$.

10. What do the quantifiers $\forall$ and $\exists$ mean?

    $\forall$ means "for all" (universal quantifier); $\exists$ means "there exists" (existential quantifier). E.g. $\forall x\, P(x)$ says $P$ holds for every $x$.

11. How do you negate the universally quantified statement $\forall x\, P(x)$?

    $\lnot \big(\forall x\, P(x)\big) \equiv \exists x\, \lnot P(x)$ — "it is not true for all $x$" becomes "there exists an $x$ for which it fails."

12. How do you negate the existentially quantified statement $\exists x\, P(x)$?

    $\lnot \big(\exists x\, P(x)\big) \equiv \forall x\, \lnot P(x)$ — "there is no $x$ with $P$" means "$P$ fails for every $x$."

13. Does the order of mixed quantifiers matter? Compare $\forall x\, \exists y$ with $\exists y\, \forall x$.

    Yes, order matters. $\forall x\, \exists y\, P(x,y)$ (a $y$ may depend on $x$) is generally weaker than $\exists y\, \forall x\, P(x,y)$ (one fixed $y$ works for all $x$).

14. Negate the statement "$\forall \varepsilon > 0,\ \exists \delta > 0$ such that $P$".

    "$\exists \varepsilon > 0$ such that $\forall \delta > 0,\ \lnot P$." Flip each quantifier and negate the inner statement.

15. What is a counterexample, and what is it used to prove?

    A counterexample is a specific instance for which a statement fails. A single counterexample disproves a universal claim $\forall x\, P(x)$ by establishing $\exists x\, \lnot P(x)$.

16. Can a counterexample ever prove a universal statement TRUE?

    No. Examples (even many) cannot prove a $\forall$ statement; they can only disprove one. Proving "for all" requires a general argument.

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

    $2$ is prime but even. So $2$ is a counterexample disproving the claim.

18. Give a counterexample to "if $x^{2} > y^{2}$ then $x > y$."

    Take $x = -3,\ y = 1$. Then $x^{2} = 9 > 1 = y^{2}$ but $x = -3 < 1 = y$. So the implication is false.

19. Give a counterexample to "$n^{2} - n + 41$ is prime for every positive integer $n$."

    $n = 41$: then $n^{2} - n + 41 = 41^{2} - 41 + 41 = 41^{2} = 1681$, which is divisible by $41$ and hence not prime.

20. What is the structure of a direct proof of $P \implies Q$?

    Assume $P$ is true, then proceed through a chain of valid logical/algebraic steps to deduce $Q$. You never assume $Q$.

21. How is an even integer and an odd integer written algebraically for use in proofs?

    An even integer is $2k$ and an odd integer is $2k+1$ for some integer $k$. These representations let you manipulate parity directly.

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

    Let the odds be $2a+1$ and $2b+1$. Their sum is $2a+1+2b+1 = 2(a+b+1)$, which is $2 \times$ an integer, hence even.

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