Binary Operations – SS1 Further Mathematics Lesson Note (First Term)

A binary operation is a rule that combines any two elements of a non-empty set to give one result.

Behavioural objectives

By the end of the lesson, students should be able to:

1. define a binary operation;
2. test whether a set is closed under a binary operation;
3. test whether a binary operation is commutative or associative;
4. test whether one operation distributes over another;
5. find the identity element and the inverse of an element under a given operation.

What is a binary operation?

A binary operation is a rule for combining any two elements of a given non-empty set.

• Symbols: binary operations are usually written with symbols such as * and Ө.
• Result: the result of combining two elements may be an element of the set, or it may not.
• Familiar operations: addition, subtraction, multiplication and division of numbers are binary operations.
• Defined operations: a question may define a new rule, such as x * y = x + y – xy, and you work with that rule only.

How do you evaluate a defined operation?

To evaluate a defined operation, replace the letters in the rule with the given values.

1. Write the rule, for example x * y = x + y – xy.
2. Replace x with the first element and y with the second element.
3. Work out the brackets and products first, then add and subtract.

What are the properties of binary operations?

There are four properties to test: closure, commutative, associative and distributive.

What is the closure property?

A non-empty set Z is closed under a binary operation * if a * b ∈ Z for all a, b ∈ Z.

• Closed: addition on the integers, because the sum of two integers is an integer.
• Not closed: subtraction on the natural numbers, because 3 – 5 = -2 is not a natural number.
• To disprove closure: one result outside the set is enough.

Example: A binary operation * is defined on the set S = {0, 1, 2, 3, 4} by x * y = x + y – xy. Find (a) 2 * 4 (b) 3 * 1 (c) 0 * 3. Is the set S closed under the operation *?

Solution

1. (a) 2 * 4 = 2 + 4 – (2 × 4) = 6 – 8 = -2
2. (b) 3 * 1 = 3 + 1 – (3 × 1) = 4 – 3 = 1
3. (c) 0 * 3 = 0 + 3 – (0 × 3) = 3
4. Since -2 ∉ S, the operation * is not closed in S.

What is the commutative property?

An operation * on a set S is commutative if a * b = b * a for all a, b ∈ S.

• Meaning: the order of combination does not affect the result.
• Commutative: addition and multiplication of real numbers.
• Not commutative: subtraction, because 5 – 3 = 2 but 3 – 5 = -2.

Steps to test it

1. Work out a * b from the rule.
2. Work out b * a by swapping the letters in the rule.
3. Simplify both sides and compare.

Example: The operation * on the set R of real numbers is defined by p * q = p³ + q³ – 3pq. Is the operation commutative?

Solution

1. p * q = p³ + q³ – 3pq
2. The commutative condition is p * q = q * p.
3. To get q * p, use the same rule but replace p by q and q by p.
4. q * p = q³ + p³ – 3qp = p³ + q³ – 3pq
5. Therefore p * q = q * p, and the operation is commutative.

What is the associative property?

An operation * on a set S is associative if (a * b) * c = a * (b * c) for all a, b, c ∈ S.

• Meaning: the way the elements are grouped does not affect the result.
• Associative: addition and multiplication of real numbers.
• Not associative: division, because (8 ÷ 4) ÷ 2 = 1 but 8 ÷ (4 ÷ 2) = 4.

Example: The operation Ө on the set Z of integers is defined by a Ө b = 2a + 3b – 1. Determine whether or not the operation is associative in Z.

Solution

1. Introduce another element c.
2. The associative condition is (a Ө b) Ө c = a Ө (b Ө c).
3. LHS: (a Ө b) Ө c = (2a + 3b – 1) Ө c = 2(2a + 3b – 1) + 3c – 1 = 4a + 6b – 2 + 3c – 1 = 4a + 6b + 3c – 3
4. RHS: a Ө (b Ө c) = a Ө (2b + 3c – 1) = 2a + 3(2b + 3c – 1) – 1 = 2a + 6b + 9c – 3 – 1 = 2a + 6b + 9c – 4
5. Since (a Ө b) Ө c ≠ a Ө (b Ө c), the operation is not associative in Z.

What is the distributive property?

One operation distributes over another when it can be shared across that other operation.

Suppose a set S is closed under two binary operations, * and Ө. For all a, b, c ∈ S:

• Left distributive: a * (b Ө c) = (a * b) Ө (a * c), so * is left distributive over Ө.
• Right distributive: (b Ө c) * a = (b * a) Ө (c * a), so * is right distributive over Ө.
• Familiar case: multiplication distributes over addition, as in 2(3 + 4) = 2 × 3 + 2 × 4.

Example: The set R of real numbers has two operations, * and Ө, defined by a * b = a + b – 3 and a Ө b = 5ab for all a, b ∈ R. Does * distribute over Ө?

Solution

1. Let a, b, c ∈ R. The condition is a * (b Ө c) = (a * b) Ө (a * c).
2. LHS: a * (b Ө c) = a * (5bc) = a + 5bc – 3
3. RHS: (a * b) Ө (a * c) = (a + b – 3) Ө (a + c – 3) = 5(a + b – 3)(a + c – 3)
4. The RHS contains a term 5a² when expanded, but the LHS has no a² term.
5. Check with a = b = c = 1: LHS = 1 + 5 – 3 = 3, and RHS = 5(-1)(-1) = 5.
6. So a * (b Ө c) ≠ (a * b) Ө (a * c), and * does not distribute over Ө.

What are identity and inverse elements?

What is an identity element?

An identity element is an element e in S such that a * e = e * a = a for all a ∈ S.

• Other name: it is also called the neutral element.
• Uniqueness: the identity element of an operation is unique.
• Addition of numbers: the identity is 0, because a + 0 = a.
• Multiplication of numbers: the identity is 1, because a × 1 = a.

Example: The operation * on the set R of real numbers is defined by a * b = (2a – 1)/2 + b for all a, b ∈ R. Determine the identity element.

Solution

1. Write the condition: a * e = e * a = a.
2. a * e = (2a – 1)/2 + e = a
3. Multiply through by 2: 2a – 1 + 2e = 2a
4. 2e = 2a – 2a + 1
5. e = ½

What is an inverse element?

The inverse of an element a is the element a^-1 such that a * a^-1 = a^-1 * a = e, where e is the identity.

• Addition of numbers: the inverse of a is -a, because a + (-a) = 0.
• Multiplication of numbers: the inverse of a is 1/a, for a ≠ 0, because a × (1/a) = 1.
• First step: always find the identity element before the inverse.

Example: An operation * is defined on the set of real numbers by x * y = x + y – 2xy. If the identity element is 0, find the inverse of the element x.

Solution

1. x * x^-1 = x^-1 * x = e, where e = 0
2. x + x^-1 – 2x x^-1 = 0
3. x^-1 – 2x x^-1 = -x
4. x^-1(1 – 2x) = -x
5. x^-1 = -x/(1 – 2x)
6. The inverse element is x^-1 = -x/(1 – 2x), which exists for x ≠ ½.

Evaluation

1. Define a binary operation.
2. State the condition for a set to be closed under a binary operation *.
3. State the condition for a binary operation to be commutative and the condition for it to be associative.
4. With x * y = x + y – xy on S = {0, 1, 2, 3, 4}, find 4 * 4. Is the result in S?
5. What is an identity element?

Assignment

1. The operation Ө on the set Z of integers is defined by a Ө b = 2a + 3b – 1. Find 2 Ө 3 and 3 Ө 2. Is the operation commutative?
2. An operation * is defined on the set of real numbers by x * y = x + y – 2xy. Show that 0 is the identity element.