Elective Mathematics SHS 1 Semester 1 Week 1: Introduction to Binary Operations (NaCCA Aligned) | Elective Mathematics SHS 1 SEM 1 WEEK 1 (WASSCE & NaCCA Aligned)

NaCCA Aligned: SHS 1, Semester 1, Week 1

Introduction to Binary Operations in Elective Mathematics

Welcome to Elective Mathematics, SHS 1! Our journey begins with Algebra, focusing on a foundational concept known as Binary Operations. Think about the basic arithmetic operations you learned in Junior High School: addition (+), subtraction (–), multiplication (×), and division (÷). Each of these takes two numbers and combines them to give exactly one result. A binary operation simply formalises this process using any defined rule.

What is a Binary Operation?

In simple terms, a binary operation is a rule that combines two elements from a set, say ‘a’ and ‘b’, to produce a unique third element, ‘c’, which must also belong to the same set. We often use special symbols like the asterisk (*), delta (Δ), or circle-plus (⊕) to denote a new operation, distinct from standard arithmetic.

The core requirement is that for any two numbers chosen from the specified set, the result of the operation must be unambiguous and must exist within that same set. For example, if we operate on the set of real numbers, the result must be a real number.

Evaluating Binary Operations

To succeed in this topic, you must master substitution. When a rule is defined, you simply replace the variables with the given values and perform the calculation strictly following the defined sequence. Let’s consider a rule defined on the set of real numbers where the operation is denoted by the asterisk (*).

If we want to evaluate 5 * 3, we substitute a=5 and b=3 into the rule:

• Step 1: Substitute a=5, b=3 into 2a + b.
• Step 2: Calculate 2(5) + 3.
• Step 3: The result is 10 + 3 = 13.

Therefore, 5 * 3 = 13. Notice how this operation is different from standard multiplication, where 5 × 3 = 15.

Ghanaian Context Example

Imagine a mobile money system where the service charge (S) for two transfers (x and y) is defined as:

If Mr. Asante transfers 500 GHS (x=500) and Mrs. Boateng transfers 100 GHS (y=100), the combined service charge is:

• 500 S 100 = (500 ÷ 10) + (100 ÷ 5)
• = 50 + 20
• = 70 GHS.

This demonstrates how binary operations model real-world rules for combining two quantities.

Complex Evaluations: Order of Operations

When solving operations that involve parentheses or brackets, we must always follow the standard mathematical order: solve the operation inside the bracket first, and then use that result for the subsequent operation. This is critical, especially when the operation is not commutative (meaning a*b is not equal to b*a).

Consider the operation defined by:

Find the value of (4 Δ 1) Δ 5.

1. Solve the Bracket (4 Δ 1):
2. Substitute x=4 and y=1: 4² + 2(1) = 16 + 2 = 18.
3. Solve the Remaining Operation (18 Δ 5):
4. Now, the first element is 18 and the second is 5.
5. Substitute x=18 and y=5: 18² + 2(5) = 324 + 10 = 334.

The final answer is 334. Skipping the bracket step would result in an incorrect answer, confirming the importance of disciplined substitution.

Working Backwards to Find Unknowns

Sometimes, we are given the rule and the final result, and we need to find one of the input values. This involves setting up an algebraic equation based on the binary rule.

Suppose the operation on the set of rational numbers is given by:

If we are told that k ⊗ 7 results in 14, what is the value of k?

We set the expression equal to the result:

Substitute p=k and q=7 into the rule:

• 3k − 7 = 14
• Add 7 to both sides: 3k = 14 + 7
• 3k = 21
• Divide by 3: k = 7

The unknown value k is 7. This reversal process tests your mastery of linear equations alongside substitution.

Constructing and Interpreting Binary Operation Tables

For finite sets (sets with a limited number of elements), we can display all possible results of a binary operation using a table, often called a Cayley table. This is extremely useful because it provides a quick visual reference for the entire set’s behaviour under that rule.

Let’s consider a small set S = {1, 2, 3} and the operation ‘Circle-Delta’ (∇) defined as standard addition, but modulo 4. Recall that modulo 4 means we only record the remainder after dividing the result by 4.

The table construction involves placing the elements of the set along the top (columns) and down the side (rows). The result of Row element ∇ Column element fills the cell at their intersection.

The full table shows all 3 × 3 = 9 possible combinations:

Analyzing these tables allows us to observe key properties, such as whether the operation is closed (all results are still within the set {0, 1, 2, 3} if we include 0 in the set for modulo arithmetic, or in the case of our original set S, we would quickly see that the result 4 is not in S={1, 2, 3}, meaning S is not closed under standard addition, but it is closed under addition modulo 4 if we slightly adjust the set to include 0). The table is the primary tool for investigating structure in abstract mathematics.

As SHS 1 students, your ability to accurately construct these tables and calculate complex algebraic substitutions is the foundation for advanced concepts in group theory and abstract algebra later in your mathematics career.

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