Binary operations are very common in WAEC and other SHS exams — especially in algebra and functions.
1. What is a Binary Operation?
A BINARY OPERATION is a rule that combines TWO ELEMENTS from a set to produce ANOTHER ELEMENT IN THE SAME SET.
Examples you already know:
Addition: 3 + 5 = 8
Multiplication: 4 \times 6 = 24
These are binary operations because:
- They combine TWO NUMBERS
- The answer is still a NUMBER.
2. Definition
A binary operation on a set S is a rule that assigns to every pair (a, b) \in S a unique element in S.
We usually write it as: a * b
Where * is the operation.
3. Examples of Binary Operations
Example 1
Let the operation * be defined on real numbers by:
a * b = a + b + 2
Find: 3 * 4
3 * 4 = 3 + 4 + 2 = 9
Example 2
If a * b = ab - 3
Find: 5 * 2 5 * 2 = (5)(2) - 3 = 10 - 3 = 7
4. Properties of Binary Operations
In SHS, you must check these properties:
(1) Closure
An operation is CLOSED if the result stays in the set.
Example:
If the set is integers, and
a * b = a + bAddition is closed because the sum of integers is an integer.
But division is NOT CLOSED on integers:
3 \div 2 = 1.5 (Not an integer)(2) Commutative Property
If: a * b = b * a
Example:
Addition is commutative:
3 + 5 = 5 + 3Check this (subtraction):
If a * b = a - b 3 * 2 = 1 2 * 3 = -1
Not equal → Not commutative
(3) Associative Property
If: (a * b) * c = a * (b * c)
Example:
Addition is associative:
(2 + 3) + 4 = 2 + (3 + 4)But subtraction is not.
(4) Identity Element
An identity element e satisfies: a * e = a
Example:
For addition, identity = 0
For multiplication, identity = 1
(5) Inverse Element
An element a^{-1} satisfies: a * a^{-1} = e
Example:
For addition, inverse of 5 is −5.
5. WAEC-Style Example
Let a * b = a + b + ab
Find the identity element.
Step 1: Let identity = e
By definition: a * e = a
Substitute into operation: a * e = a + e + ae
Set equal to a: a + e + ae = a
Step 2: Simplify
Subtract a: e + ae = 0
Factor: e(1 + a) = 0
For all a, this is only possible if: e = 0
Final Answer:
Identity element = 0
6. Common WAEC Traps ⚠️
1. Forgetting to substitute correctly.
2. Mixing normal arithmetic with defined operation.
3. Forgetting that identity must work for all values of a.
4. Not checking closure.
5. Expanding brackets wrongly.
7. Summary Table
| Property | What to Check |
| Closure | Result still in the set ? |
| Commutative | a * b = b * a ? |
| Associative | (a * b) * c = a * (b * c) ? |
| Identity | Does a * e = a ? |
| Inverse | Does a * a^{-1} = e ? |