LET Secondary Mathematics — Logic and Mathematics in the Modern WorldRevision Notes

Formulas

Exam Tips

• Memorize the ONE row that makes the conditional FALSE: p = T, q = F → p → q = F.
• For truth-table questions, build the table systematically: fill in p and q columns first, then compute ¬p and ¬q, then compound expressions left to right.
• When a question asks for the truth value of a long compound statement, substitute T/F for each variable and simplify step by step.
• Remember: conjunction (∧) is strict — it needs BOTH; disjunction (∨) is lenient — it needs ONLY ONE.

Key Points

• A statement (proposition) is a declarative sentence that is either TRUE or FALSE — never both and never neither.
• Commands ('Close the door.') and questions ('Is Manila the capital?') are NOT statements because they cannot be assigned a definite truth value.
• Simple statements express one idea; compound statements join two or more simple statements using logical connectives.
• Negation (¬p) flips the truth value: if p is TRUE, then ¬p is FALSE, and vice versa.
• Conjunction (p ∧ q) is TRUE only when BOTH p and q are TRUE — if even one is false, the conjunction is false.
• Disjunction (p ∨ q) is TRUE when AT LEAST ONE of p or q is TRUE — it is false only when both are false.
• Conditional (p → q) is FALSE only when the hypothesis (p) is TRUE and the conclusion (q) is FALSE. All other combinations are true.
• Biconditional (p ↔ q) is TRUE when p and q have the SAME truth value (both true or both false).
• The most tested truth-value row in the LET: a conditional p → q is FALSE when p = TRUE and q = FALSE.

Definitions

Section Title

Statements and Logical Connectives

Common Mistakes

• Thinking a conditional is false when the hypothesis is false — it is actually TRUE (a false hypothesis makes the conditional vacuously true).
• Confusing inclusive 'or' (at least one true = TRUE) with exclusive 'or' (exactly one true). Mathematics uses inclusive or.
• Labeling commands and questions as statements — they are NOT propositions because you cannot assign them a truth value.
• Forgetting that negation applies to the WHOLE statement, not just the subject noun.

Formulas

Exam Tips

• Use the memory aid 'CONIC': Converse = O-pposite order; Inverse = N-egate; Contrapositive = C-ombine both.
• Quick check: original and contrapositive always match in truth value; converse and inverse always match each other.
• For De Morgan's: 'NOT (A and B)' = 'NOT A OR NOT B'. Change the connective AND flip each part.
• If a question asks which statement is logically equivalent to a given conditional, the answer is its contrapositive.

Key Points

• From any conditional p → q, three related forms can be derived: converse, inverse, and contrapositive.
• CONVERSE: q → p — swap the hypothesis and conclusion.
• INVERSE: ¬p → ¬q — negate both the hypothesis and conclusion.
• CONTRAPOSITIVE: ¬q → ¬p — swap AND negate both parts.
• The CONTRAPOSITIVE is LOGICALLY EQUIVALENT to the original conditional — they always have the same truth value.
• The CONVERSE and INVERSE are logically equivalent to EACH OTHER but NOT to the original conditional.
• Two statements are logically equivalent (≡) if they have identical truth values in every possible scenario.
• A TAUTOLOGY is a compound statement that is ALWAYS TRUE regardless of the truth values of its components.
• A CONTRADICTION is a compound statement that is ALWAYS FALSE regardless of the truth values of its components.

Definitions

Section Title

Variations of the Conditional and Logical Equivalence

Common Mistakes

• Treating the converse as logically equivalent to the original — it is NOT. Only the contrapositive is equivalent.
• Confusing 'inverse' and 'contrapositive' — the inverse negates both parts WITHOUT swapping; the contrapositive does BOTH.
• Applying De Morgan's Law incorrectly by only negating one part of the compound statement.
• Forgetting that the converse and inverse are equivalent to EACH OTHER (not to the original).

Formulas

Exam Tips

• On LET multiple-choice, identify the argument form first: is premise 2 affirming p (modus ponens) or denying q (modus tollens)? Both are valid.
• If premise 2 affirms q or denies p, the argument is a FALLACY — mark it as invalid.
• For inductive/deductive questions: if the question gives specific examples and draws a general rule, it is INDUCTIVE. If it applies a general rule to a specific case, it is DEDUCTIVE.
• Remember: in K–12 BEC, the inquiry-based approach starts inductively (pupils observe → discover pattern) then moves deductively (teacher formalizes the rule).

Key Points

• An argument is VALID if, whenever all the premises are true, the conclusion must also be true.
• MODUS PONENS (affirming the antecedent): Premise 1: p → q. Premise 2: p. Conclusion: Therefore, q. VALID.
• MODUS TOLLENS (denying the consequent): Premise 1: p → q. Premise 2: ¬q. Conclusion: Therefore, ¬p. VALID.
• FALLACY — Affirming the Consequent: Premise 1: p → q. Premise 2: q. Conclusion: p. INVALID.
• FALLACY — Denying the Antecedent: Premise 1: p → q. Premise 2: ¬p. Conclusion: ¬q. INVALID.
• INDUCTIVE REASONING: moves from specific observations to a general conclusion (probable, not certain). One counterexample disproves it.
• DEDUCTIVE REASONING: moves from general premises to a specific, certain conclusion. If premises are true and logic is valid, the conclusion MUST be true.
• Teachers use INDUCTIVE reasoning to let pupils discover patterns; DEDUCTIVE reasoning to formally prove rules.

Definitions

Section Title

Valid Arguments: Modus Ponens and Modus Tollens

Common Mistakes

• Treating 'affirming the consequent' as valid — knowing q is true does NOT prove p is true.
• Treating 'denying the antecedent' as valid — knowing p is false does NOT prove q is false.
• Confusing inductive and deductive reasoning — inductive gives probable conclusions; deductive gives certain ones.
• Thinking a valid argument guarantees true conclusions — validity is about logical FORM; truth of premises is a separate issue.

Formulas

Exam Tips

• IDENTIFY FIRST: subtract consecutive terms — if the differences are constant, it is arithmetic; divide consecutive terms — if ratios are constant, it is geometric; if each term equals the sum of the two before it, it is Fibonacci.
• For 'which term equals a given value' problems: set the nth-term formula equal to that value and solve for n algebraically.
• For sum problems: use Sₙ = (n/2)(a₁ + aₙ). Always find aₙ first using the nth-term formula before summing.
• Know the first 10 Fibonacci numbers by heart: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
• Golden ratio quick fact for MCQ: consecutive Fibonacci ratio → 55/34 ≈ 1.618.

Key Points

• A SEQUENCE is an ordered list of numbers (terms) that follow a specific rule.
• ARITHMETIC SEQUENCE: a constant value called the COMMON DIFFERENCE (d) is added to each term to get the next term.
• To find d in an arithmetic sequence: d = aₙ₊₁ − aₙ (subtract any term from the next term).
• GEOMETRIC SEQUENCE: a constant value called the COMMON RATIO (r) is multiplied to each term to get the next term.
• To find r in a geometric sequence: r = aₙ₊₁ / aₙ (divide any term by the previous term).
• FIBONACCI SEQUENCE: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... Each term is the sum of the TWO preceding terms.
• The ratio of consecutive Fibonacci numbers approaches the GOLDEN RATIO φ ≈ 1.618 (phi).
• The Fibonacci sequence and golden ratio appear in nature: flower petals, pineapple spirals, nautilus shells, and sunflower seeds.
• SQUARE NUMBERS: 1, 4, 9, 16, 25, 36, ... (nth term = n²)
• TRIANGULAR NUMBERS: 1, 3, 6, 10, 15, 21, ... (nth term = n(n+1)/2)
• To find a specific term, FIRST identify whether the sequence is arithmetic, geometric, or Fibonacci, THEN apply the correct formula.

Definitions

Section Title

Patterns, Sequences, and the Fibonacci Sequence

Common Mistakes

• Using the geometric formula for an arithmetic sequence (and vice versa) — ALWAYS check: is it adding or multiplying?
• Making an off-by-one error in the formula aₙ = a₁ + (n−1)d — the exponent/multiplier is (n−1), NOT n.
• Confusing the sum formula Sₙ with the nth-term formula — Sₙ adds up ALL terms; aₙ finds ONE specific term.
• In geometric sequences, calculating r^n instead of r^(n−1) in the nth-term formula.
• Misidentifying a Fibonacci-type sequence as arithmetic or geometric — always check if consecutive differences and ratios are constant.

Formulas

Exam Tips

• Draw a Venn diagram for every set-counting problem — fill in the INTERSECTION first, then subtract to find the 'only A' and 'only B' regions.
• Remember: the total in the Venn diagram (all regions including outside) = Universal set size.
• Connect sets to logic for efficiency: if a question about sets uses 'and,' think intersection; 'or,' think union; 'not,' think complement.
• For three-set problems, use: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C).

Key Points

• A SET is a well-defined collection of distinct objects called ELEMENTS.
• UNION (A ∪ B): All elements in A OR B (or both). Corresponds to logical disjunction (∨).
• INTERSECTION (A ∩ B): Only elements in BOTH A AND B. Corresponds to logical conjunction (∧).
• COMPLEMENT (A'): All elements NOT in A (from the universal set). Corresponds to logical negation (¬).
• SUBSET: A ⊆ B means every element of A is also in B.
• EMPTY SET (∅): A set with no elements. It is a subset of every set.
• INCLUSION-EXCLUSION PRINCIPLE: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Subtract the overlap to avoid double-counting.
• Venn diagrams visually show relationships between sets and are used to solve counting problems.
• IMPORTANT PARALLEL: Set Union ↔ Logical OR; Set Intersection ↔ Logical AND; Set Complement ↔ Logical NOT.

Definitions

Section Title

Sets and Venn Diagrams

Common Mistakes

• Forgetting to subtract the intersection in the inclusion-exclusion formula, causing double-counting.
• Confusing union (∪) and intersection (∩) — union is BIGGER (more elements); intersection is SMALLER (only shared elements).
• Placing elements that belong to ONLY A inside the intersection region of the Venn diagram.
• Forgetting to account for 'neither' category — subtract the union total from the universal set total.

Formulas

Exam Tips

• For comparison questions: simple interest total = P + PRT = P(1 + RT); compound interest total = P(1 + r)ⁿ. Compound always yields MORE for the same rate and time (beyond 1 period).
• Rate conversion: always convert percent to decimal before substituting into any formula (8% → 0.08; 5% → 0.05).
• For check-digit conceptual questions: know that a mismatch in the checksum means an error was detected — you do not need to compute ISBNs from scratch.
• Data interpretation questions (reading bar graphs, pie charts, tables) test whether you can extract and compute from presented information — practice quickly reading graph labels and scales.

Key Points

• SIMPLE INTEREST: Interest is calculated only on the original principal. The amount grows LINEARLY.
• COMPOUND INTEREST: Interest is calculated on the principal AND on accumulated interest. The amount grows EXPONENTIALLY (faster than simple interest).
• In compound interest, the key phrase is 'interest earns interest' — this is why it grows faster over time.
• MODULAR ARITHMETIC: Working with remainders after division. The notation a ≡ b (mod n) means a and b have the same remainder when divided by n.
• CHECK DIGITS on barcodes, ISBNs, and bank cards use modular arithmetic to detect single-digit errors. A wrong digit produces a checksum mismatch.
• STATISTICS and DATA: Mean, median, mode, and range describe data sets. Graphs (bar, line, pie) translate data into visual information for decision-making.
• Mathematics is a PRACTICAL LITERACY for daily life: budgeting a salary, reading a utility bill, understanding loan terms, and interpreting health data all require mathematical reasoning.
• As elementary teachers, financial literacy topics link to real-life applications in the K–12 curriculum (e.g., Grade 5–6 Math competencies on money and percentage).

Definitions

Section Title

Mathematics in Finance, Codes, and Data

Common Mistakes

• Using the rate R as a percentage instead of converting to a decimal — 8% must become 0.08 in formulas.
• Confusing the simple interest I (interest only) with the total amount A = P + I.
• Using T = months instead of converting to years in the simple interest formula — always express time in YEARS unless told otherwise.
• In compound interest, computing A = P × r × n instead of A = P(1 + r)ⁿ — the base is (1 + r), not r alone.