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

Mathematical Reasoning and Proof syllabus — full chapter and topic list

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

1. 1

   Logic and Argument Structure

   3 topics

   • Statements and Truth Values
     • Propositions and negation
     • Conjunction and disjunction
   • Implication and Equivalence
     • If-then statements
     • Converse, inverse and contrapositive
     • Necessary and sufficient conditions
   • Quantifiers
     • For all and there exists
     • Negating quantified statements
2. 2

   Methods of Proof

   4 topics

   • Direct Proof
     • Deductive chains of reasoning
   • Proof by Exhaustion
     • Checking all cases
   • Proof by Contradiction
     • Assuming the negation
     • Classic irrationality and infinitude proofs
   • Disproof by Counterexample
     • Constructing a single counterexample
3. 3

   Analysing Mathematical Arguments

   3 topics

   • Identifying Flawed Reasoning
     • Spotting invalid algebraic steps
     • Division by zero and sign errors
   • Validity versus Conclusion
     • Correct answer from incorrect reasoning
     • Gaps and unjustified leaps
   • Generalisation and Special Cases
     • When a result extends and when it fails
4. 4

   Inference and Conditions

   3 topics

   • Deductions from Given Information
     • Drawing valid conclusions
     • What cannot be concluded
   • Conditions on Equations and Inequalities
     • Existence conditions for solutions
     • Constraints on parameters
   • Reasoning with Sets of Solutions
     • Intersection and union of solution sets

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

25 of 51 cards from the Mathematical Reasoning and Proof deck — real questions with worked answers.

1. What is a mathematical statement (proposition)?

   A declarative sentence that is either definitively true or definitively false, but not both. For example, $7$ is prime is a statement; $x > 3$ is not (it depends on $x$) until $x$ is fixed.

2. What is the truth value of a statement?

   It is whether the statement is true (T) or false (F). Every well-formed proposition has exactly one truth value.

3. What is the negation of a statement $P$, and how does its truth value relate to $P$?

   The negation $\neg P$ (not $P$) is true exactly when $P$ is false and false exactly when $P$ is true. So $\neg P$ always has the opposite truth value to $P$.

4. Give the truth table for the conjunction $P \land Q$ ("$P$ and $Q$").

   $P \land Q$ is true only when both $P$ and $Q$ are true; it is false in all other cases (TT$\to$T, TF$\to$F, FT$\to$F, FF$\to$F).

5. Give the truth table for the disjunction $P \lor Q$ ("$P$ or $Q$").

   The inclusive "or" $P \lor Q$ is true when at least one of $P$, $Q$ is true; it is false only when both are false (TT$\to$T, TF$\to$T, FT$\to$T, FF$\to$F).

6. State De Morgan's laws for negating compound statements.

   $\neg(P \land Q) \equiv \neg P \lor \neg Q$ and $\neg(P \lor Q) \equiv \neg P \land \neg Q$. Negation swaps "and" with "or" and negates each part.

7. What does the implication $P \Rightarrow Q$ mean, and when is it false?

   It means "if $P$ then $Q$": $P$ is sufficient for $Q$, and $Q$ is necessary for $P$. It is false in exactly one case — when $P$ is true and $Q$ is false.

8. Complete the truth table for $P \Rightarrow Q$.

   TT$\to$T, TF$\to$F, FT$\to$T, FF$\to$T. An implication with a false antecedent (premise) is vacuously true.

9. In $P \Rightarrow Q$, name $P$ and $Q$ and state the necessary/sufficient relationship.

   $P$ is the antecedent (hypothesis), $Q$ is the consequent (conclusion). $P$ is a sufficient condition for $Q$; $Q$ is a necessary condition for $P$.

10. Define the converse, inverse and contrapositive of $P \Rightarrow Q$.

    Converse: $Q \Rightarrow P$. Inverse: $\neg P \Rightarrow \neg Q$. Contrapositive: $\neg Q \Rightarrow \neg P$.

11. Which of converse, inverse and contrapositive is logically equivalent to $P \Rightarrow Q$?

    The contrapositive $\neg Q \Rightarrow \neg P$ is logically equivalent to $P \Rightarrow Q$. The converse and inverse are equivalent to each other but NOT to the original.

12. What is the logical equivalence $P \Leftrightarrow Q$ (biconditional)?

    "$P$ if and only if $Q$" (iff): it is true when $P$ and $Q$ have the same truth value, false when they differ. It means both $P \Rightarrow Q$ and $Q \Rightarrow P$ hold.

13. How do you write the negation of the implication $P \Rightarrow Q$?

    $\neg(P \Rightarrow Q) \equiv P \land \neg Q$. The implication fails precisely when the hypothesis holds but the conclusion does not.

14. Express $P \Rightarrow Q$ using only $\lor$ and $\neg$.

    $P \Rightarrow Q \equiv \neg P \lor Q$.

15. What is the universal quantifier and how is it written?

    "For all" / "for every", written $\forall$. The statement $\forall x \, P(x)$ asserts $P(x)$ is true for every $x$ in the domain.

16. What is the existential quantifier and how is it written?

    "There exists" / "for some", written $\exists$. The statement $\exists x \, P(x)$ asserts $P(x)$ is true for at least one $x$ in the domain.

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

    $\neg\big(\forall x \, P(x)\big) \equiv \exists x \, \neg P(x)$. "Not all" becomes "there exists one that doesn't."

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

    $\neg\big(\exists x \, P(x)\big) \equiv \forall x \, \neg P(x)$. "None exists" means "all fail."

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

    Yes, order matters. $\forall x \, \exists y \, P(x,y)$ allows $y$ to depend on $x$, while $\exists y \, \forall x \, P(x,y)$ requires a single $y$ that works for every $x$. The latter is stronger.

20. What is a direct proof of $P \Rightarrow Q$?

    You assume $P$ is true and, through a chain of valid logical/algebraic deductions, derive $Q$. No appeal to contradiction or cases is needed.

21. 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. $\blacksquare$

22. 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.

23. 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.

24. 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.

25. 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 $n$.

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