Further Mathematics · Lesson 2

Binary operations & mathematical logic

WAECNECO

What the syllabus asks

WAEC: binary operations (closure, commutativity, associativity, distributivity, identity and inverse elements; operation tables); mathematical logic (simple and compound statements, negation, conjunction, disjunction, implication, biconditional, truth tables, converse, inverse, contrapositive, tautology and contradiction, logical equivalence, De Morgan’s laws)

NECO: binary operations (closure, commutativity, associativity, distributivity, identity and inverse elements; operation tables); mathematical logic (simple and compound statements, negation, conjunction, disjunction, implication, biconditional, truth tables, converse, inverse, contrapositive, tautology and contradiction, logical equivalence, De Morgan’s laws)

What you will learn

Evaluate expressions involving defined binary operations.
Test a binary operation for closure, commutativity, associativity and distributivity.
Find the identity element and inverses for an operation.
Construct and read operation tables on finite sets, including modular arithmetic.
Identify propositions and form compound statements with logical connectives.
Construct truth tables and recognise tautologies, contradictions and equivalences.
Write the converse, inverse and contrapositive of an implication and use De Morgan’s laws.

Key points

A binary operation combines two elements of a set into one; substitute carefully in order.
Closure: results stay in the set; one counter-example disproves.
Commutative: a∗b = b∗a; associative: (a∗b)∗c = a∗(b∗c); prove algebraically.
Identity e: solve a∗e = a for a constant e; check both sides.
Inverse: solve a∗a⁻¹ = e; note elements with no inverse.
Operation tables: symmetry → commutative; row equal to heading → identity; locate e in each row for inverses.
Distributive: a∗(b∘c) = (a∗b)∘(a∗c).
Connectives: ~, ∧, ∨, ⇒ (false only T⇒F), ⇔ (true when values match).
Contrapositive ~q ⇒ ~p is equivalent to p ⇒ q; converse and inverse are not.
Tautology always true; contradiction always false; De Morgan: ~(p∧q) ≡ ~p∨~q, ~(p∨q) ≡ ~p∧~q.

Start here: What a binary operation is

Addition takes two numbers and gives one number: 3 + 4 = 7. Multiplication does the same: 3 × 4 = 12. Each is a binary operation ("binary" means two inputs). In Further Maths, examiners invent their own operations with symbols such as ∗, ∘, ⊕, Δ and ask you to work with them.

A binary operation ∗ on a set S is a rule that assigns to every ordered pair (a, b) of elements of S a single element a ∗ b.

The rule is given by a formula. Your job is to substitute carefully: the first number replaces a, the second replaces b.

Worked example 2.1

The operation ∗ is defined on ℝ by a ∗ b = a + b + 2ab. Evaluate (a) 3 ∗ 4, (b) (−1) ∗ 5, (c) ½ ∗ (−2).

(a) 3 + 4 + 2(3)(4) = 7 + 24 = 31.
(b) −1 + 5 + 2(−1)(5) = 4 − 10 = −6.
(c) ½ + (−2) + 2(½)(−2) = −1.5 − 2 = −3.5.

31; −6; −3.5

Worked example 2.2 (nested operations)

a Δ b = a² − b. Find (2 Δ 3) Δ 4 and 2 Δ (3 Δ 4).

2 Δ 3 = 4 − 3 = 1; then 1 Δ 4 = 1 − 4 = −3.
3 Δ 4 = 9 − 4 = 5; then 2 Δ 5 = 4 − 5 = −1.
Work inside the brackets first, exactly as in ordinary arithmetic.

−3 and −1

Worked example 2.3 (solving)

On ℝ, x ∘ y = 2x + y − 3. Solve x ∘ 5 = 14.

Substitute: 2x + 5 − 3 = 14.
2x = 12, x = 6.

x = 6

⚠️ Common mistake: swapping a and b. For a Δ b = a² − b, 3 Δ 2 = 9 − 2 = 7, but 2 Δ 3 = 4 − 3 = 1. Order matters unless the operation is commutative.

Try it yourself. If p ⊕ q = pq − p + q, find 4 ⊕ (−2).

Show solution

4(−2) − 4 + (−2) = −8 − 4 − 2 = −14.

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Practice questions

1. If a ∗ b = 3a + 2b, find 2 ∗ (−1) and (−1) ∗ 2.

Show worked answer

4 and 1.

2. Find the identity for a ∘ b = a + b + 7 on ℝ.

Show worked answer

e = −7.

3. For a ∘ b = a + b + 7, find the inverse of 3.

Show worked answer

3 + x + 7 = −7 → x = −17.

4. Is a ∗ b = 2a + 2b associative? Explain.

Show worked answer

(a∗b)∗c = 4a + 4b + 2c; a∗(b∗c) = 2a + 4b + 4c: not associative.

5. In multiplication mod 7 on {1,…,6}, find the inverse of 3.

Show worked answer

3 × 5 = 15 = 1 (mod 7): 5.

Key words

Binary operation: A rule combining two elements of a set to give a single element.
Closure: The result of the operation is always in the set.
Commutative: The order does not matter: a∗b = b∗a.
Associative: Grouping does not matter: (a∗b)∗c = a∗(b∗c).
Distributive: One operation spreads over another, as × over +.
Identity element: The element e with a∗e = e∗a = a for every a.
Inverse element: The element a⁻¹ with a∗a⁻¹ = a⁻¹∗a = e.
Operation table: A grid (Cayley table) showing all results of a binary operation on a finite set.
Proposition: A statement that is either true or false, not both.
Negation: The statement "not p", written ~p.
Conjunction: "p and q", p ∧ q; true only when both are true.
Disjunction: "p or q", p ∨ q; true when at least one is true.

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