Mathematics · Lesson 7

Binary operations, modular arithmetic & logical reasoning

JAMBWAECNECO

What the syllabus asks

JAMB: Number and numeration: modular arithmetic; Algebra: binary operations; Sets and logic: logical reasoning

WAEC: Number and numeration: modular arithmetic (addition, subtraction, multiplication, solving simple equations, application to daily life); binary operations (closure, commutativity, associativity, identity, inverse, distributivity); logical reasoning (simple and compound statements, negation, truth tables, implication, valid arguments)

NECO: Number and numeration: modular arithmetic (addition, subtraction, multiplication, solving simple equations, application to daily life); binary operations (closure, commutativity, associativity, identity, inverse, distributivity); logical reasoning (simple and compound statements, negation, truth tables, implication, valid arguments)

What you will learn

Reduce numbers modulo n and perform modular addition, subtraction and multiplication.
Solve simple modular equations and apply them to days, clocks and cycles.
Evaluate binary operations defined by formulas and by tables.
Test closure, commutativity, associativity and distributivity.
Find identity elements and inverses.
Write truth tables, form converse and contrapositive, and judge simple arguments.

Key points

a mod n is the remainder 0 to n − 1; add n to negative numbers to bring them into range.
Calculate normally, then reduce; reduce powers step by step.
Solve modular equations by testing every value from 0 to n − 1.
Binary operation: substitute exactly in order; do inner brackets first.
Closure, commutativity, associativity: prove with algebra, disprove with one counterexample.
Identity: solve a ∗ e = a for a constant e; inverse: solve a ∗ b = e.
In tables: identity row repeats the header; symmetry means commutative.
Logic: p → q false only for T → F; equivalent to its contrapositive, not its converse.

Start here: Remainders and the meaning of mod

A clock face resets after 12. If it is 9 o’clock now, 5 hours later it is 2 o’clock, not 14 o’clock. Days of the week reset after 7. This "wrap-around" counting is modular arithmetic.

a mod n = the remainder when a is divided by n (0 ≤ remainder < n).
a ≡ b (mod n) means a and b leave the same remainder, i.e. n divides a − b.

Worked example 7.1

Find (a) 47 mod 5; (b) 23 mod 7; (c) 60 mod 12.

(a) 47 = 9 × 5 + 2 → 2.
(b) 23 = 3 × 7 + 2 → 2.
(c) 60 = 5 × 12 + 0 → 0.

2; 2; 0

Worked example 7.2 (negative numbers)

Find −3 mod 7.

We need a remainder between 0 and 6.
Add 7 until it is in range: −3 + 7 = 4.
Check: −3 = (−1) × 7 + 4 ✓.

So in mod 7, the "numbers" are just 0, 1, 2, 3, 4, 5, 6, and every integer is equivalent to one of them.

⚠️ Common mistake: giving a negative answer (−3) or an answer equal to n (7). In mod 7, answers must be from 0 to 6.

Try it yourself. Find 125 mod 9.

Show solution

125 = 13 × 9 + 8 → 8.

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Practice questions

1. Find 1000 mod 7.

Show worked answer

1000 = 142 × 7 + 6 → 6.

2. Solve 4x ≡ 3 (mod 5).

Show worked answer

4 × 2 = 8 ≡ 3: x = 2.

3. Today is Tuesday. What day is it after 45 days?

Show worked answer

45 mod 7 = 3: Friday.

4. a ∗ b = 3a − 2b. Find (2 ∗ 1) ∗ 3.

Show worked answer

2 ∗ 1 = 4; 4 ∗ 3 = 12 − 6 = 6.

5. For a ∗ b = a + b − 5, find the identity and the inverse of 2.

Show worked answer

e = 5; 2 + b − 5 = 5 → b = 8.

Key words

Modulo (mod): The remainder after division by a fixed number n.
Congruent: a ≡ b (mod n) means a and b leave the same remainder on division by n.
Binary operation: A rule combining two elements of a set to give one element.
Closure: The result of the operation is always in the set.
Commutative: Order does not matter: a∗b = b∗a.
Associative: Grouping does not matter: (a∗b)∗c = a∗(b∗c).
Identity element: e such that a∗e = e∗a = a for all a.
Inverse: a⁻¹ such that a∗a⁻¹ = a⁻¹∗a = e.
Statement (proposition): A sentence that is either true or false, not both.
Negation: ~p: "not p", the opposite truth value.
Conjunction / disjunction: p ∧ q (and) / p ∨ q (or).
Contrapositive: For p → q, the statement ~q → ~p.

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