Exam Prep: Prelims & Finals

Your prelim in Mathematics in the Modern World is built almost entirely from the first three units of this subject. If you can do everything on this page, you can pass the prelim. Here is exactly what to expect and how to prepare for it in one week.

What a Typical Prelim Covers

Typical Question-Type Breakdown

• Multiple choice (~35–40%): fast definition and classification items. These are free points if you memorized the lists below.
• True or false (~20–25%): almost always aimed at the classic confusions — converse vs. contrapositive, inductive vs. deductive, inclusive "or".
• Computations and constructions (~35–45%): truth tables, set operations, a Venn diagram word problem, Fibonacci calculations, and one Polya-style word problem. This is where exams are won or lost, so most of your practice time should go here.

The Exact Memorize List

Everything below appears repeatedly on prelims. Write this on one sheet and drill it until it is automatic.

Unit I

• Fibonacci definition: $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Know the first twelve terms cold: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144$.
• Golden ratio: $\varphi = \dfrac{1 + \sqrt{5}}{2} \approx 1.618$, and its conjugate $\bar{\varphi} = \dfrac{1 - \sqrt{5}}{2} \approx -0.618$.
• Binet's formula: $F_n = \dfrac{\varphi^n - \bar{\varphi}^n}{\sqrt{5}}$.
• Symmetry types: bilateral (one mirror axis) vs. radial (repeats around a center).
• Rosette patterns: cyclic $C_n$ (rotations only) vs. dihedral $D_n$ (rotations and reflections).
• The seven Conway frieze names: hop, step, sidle, spinning hop, spinning sidle, jump, spinning jump — and which symmetries each allows.
• Exactly 7 frieze groups and exactly 17 wallpaper pattern types exist.

Unit II

• Truth table rules in one line each: $p \wedge q$ is true only when both are true; $p \vee q$ is false only when both are false; $p \longrightarrow q$ is false only when a true premise meets a false conclusion; $p \longleftrightarrow q$ is true when both values match.
• A false premise makes any conditional true by default.
• From $p \longrightarrow q$: converse is $q \longrightarrow p$, inverse is $\neg p \longrightarrow \neg q$, contrapositive is $\neg q \longrightarrow \neg p$. Only the contrapositive is equivalent to the original. The converse and inverse are equivalent to each other.
• De Morgan's laws: $\neg(p \vee q) \iff (\neg p \wedge \neg q)$ and $\neg(p \wedge q) \iff (\neg p \vee \neg q)$.
• Tautology (all rows true), contradiction (all rows false), contingency (mixed).
• Set operations: $A \cup B$ (either), $A \cap B$ (both), $A \setminus B$ (in $A$ but not $B$), $A'$ (everything in $U$ outside $A$). Cardinality $n(A)$ counts distinct elements.
• Venn counting for two sets: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. For three-set word problems, always fill the diagram starting from the center (all three) outward.

Unit III

• Inductive reasoning: specific observations $\to$ general conjecture (never guaranteed; one counterexample kills it). Deductive reasoning: general principles $\to$ specific certainty.
• Polya's four steps, in order: (1) understand the problem, (2) devise a plan, (3) execute the plan, (4) review and reflect.
• Handshake-type counting: $n$ people each shaking hands once means $\dfrac{n(n-1)}{2}$ handshakes.

Top Mistakes Students Actually Make

1. Treating the converse as equivalent to the original conditional. It is not — only the contrapositive is.
2. Marking $p \vee q$ false when both parts are true. The "or" in logic is inclusive: both true means the disjunction is true.
3. Forgetting that a false premise makes the whole conditional true, and marking $F \longrightarrow F$ as false.
4. Off-by-one Fibonacci indexing: the 10th term is 55, not 89. Count carefully from $F_1 = 1$.
5. Adding $n(A) + n(B)$ in Venn problems without subtracting the overlap — double-counting the intersection.
6. Writing $x \subseteq A$ when they mean $x \in A$. Elements belong with $\in$; only sets sit inside $\subseteq$.
7. "Proving" a conjecture with three examples. Inductive evidence is not proof; a single counterexample disproves it.

A Realistic 7-Day Study Plan

Work through these under exam conditions first — 25 minutes, no notes — then check the key. Every item is drawn straight from Units I–III.

Items

1. The Fibonacci sequence begins $1, 1, 2, 3, 5, 8, 13, 21, \dots$ Write the next two terms.

2. Which of the following exhibits radial (rotational) symmetry? (A) a butterfly (B) a snowflake (C) a human face (D) a swan

3. Given $F_{14} = 377$ and $F_{16} = 987$, determine $F_{15}$.

4. Exactly how many distinct frieze pattern groups exist, and exactly how many distinct wallpaper pattern types exist?

5. Write the exact expression for the golden ratio $\varphi$ and its approximate decimal value.

6. Decide whether each sentence is a proposition. If it is, give its truth value.

• (a) "Manila is the capital of the Philippines."
• (b) "Please submit your homework."
• (c) "Is 91 a prime number?"
• (d) "$15 - 6 = 8$."

7. Suppose $p$ is true and $q$ is false. Evaluate: (a) $p \wedge q$ (b) $p \vee q$ (c) $p \longrightarrow q$ (d) $p \longleftrightarrow q$

8. Write the contrapositive of: "If it rains, then the class is suspended."

9. Use De Morgan's law to write the negation of: "The number is even or the number is positive."

10. Let $U = \{1,2,3,4,5,6,7,8\}$, $A = \{1,2,3,4\}$, $B = \{3,4,5,6\}$. Find: (a) $A \cap B$ (b) $A \cup B$ (c) $(A \cup B)'$ (d) $A \setminus B$

11. Using the same set $A = \{1,2,3,4\}$: (a) state $n(A)$; (b) true or false: $\{1,2\} \subseteq A$.

12. Construct the truth table for $p \vee \neg p$ and classify the proposition as a tautology, contradiction, or contingency.

13. Classify each argument as inductive or deductive:

• (a) "Our last three quizzes each had a Venn diagram problem, so the prelim will have one too."
• (b) "Every multiple of 10 ends in 0. The number 570 is a multiple of 10. Therefore 570 ends in 0."

14. At a meeting, 10 people each shake hands with every other person exactly once. How many handshakes occur?

15. A number is doubled and then increased by 6, giving 20. Working backwards, find the number.

Answer Key

1. $34$ and $55$. Each term is the sum of the previous two: $13 + 21 = 34$, then $21 + 34 = 55$.

2. (B) a snowflake. It repeats around a central point. The butterfly, face, and swan are the textbook bilateral (mirror) examples.

3. Since $F_{16} = F_{15} + F_{14}$, rearrange: $F_{15} = F_{16} - F_{14} = 987 - 377 = \mathbf{610}$.

4. Exactly 7 frieze groups (Conway's hop through spinning jump) and exactly 17 wallpaper types.

5. $\varphi = \dfrac{1 + \sqrt{5}}{2} \approx 1.618$.

6. (a) Proposition, true. (b) Not a proposition — imperative. (c) Not a proposition — interrogative. (d) Proposition, false (the equation fails, but it still has a definite truth value, so it qualifies).

7. (a) $p \wedge q$ is false (needs both true). (b) $p \vee q$ is true (one true is enough). (c) $p \longrightarrow q$ is false (true premise, false conclusion — the only failing row). (d) $p \longleftrightarrow q$ is false (values differ).

8. Swap and negate both parts: "If the class is not suspended, then it does not rain." (Order matters: negated conclusion first.)

9. $\neg(p \vee q) \iff (\neg p \wedge \neg q)$: "The number is not even and the number is not positive."

10. (a) $A \cap B = \{3,4\}$ (b) $A \cup B = \{1,2,3,4,5,6\}$ (c) $(A \cup B)' = \{7,8\}$ (d) $A \setminus B = \{1,2\}$.

11. (a) $n(A) = 4$. (b) True — every element of $\{1,2\}$ is in $A$.

12. | $p$ | $\neg p$ | $p \vee \neg p$ | |---|---|---| | 1 | 0 | 1 | | 0 | 1 | 1 |

True in every row, so it is a tautology.

13. (a) Inductive — a general expectation drawn from a few specific observations, and not guaranteed. (b) Deductive — a general rule applied to a specific case, giving certainty.

14. Each of the 10 people shakes 9 hands, but that counts every handshake twice: $\dfrac{10 \times 9}{2} = \mathbf{45}$.

15. Reverse the operations in reverse order: $20 - 6 = 14$, then $14 \div 2 = \mathbf{7}$. Check forward: $7 \times 2 + 6 = 20$. ✓

Scored below 12? Go back to the blueprint's memorize list before attempting a full mock. The four full 30-item mock exams below, with completely worked answer keys, come with the subject unlock.