BINARY OPERATIONS

WEEK SIX

SS1 FURTHER MATHS FIRST TERM

BINARY OPERATIONS

CONTENT

Definition of Binary Operation

Properties of Binary Operations: Closure, Commutative law, Associative law, Distributive law,

Laws of Complementation as in a Set: Identity Elements and Inverse of an Elements

Multiplication tables of Binary Operations

SUB TOPIC: DEFINITION OF BINARY OPERATIONS

A binary of operation is any rule of combination of any two elements of a given non-empty set. Asterisk symbol is used to denote binary operation. Some authors uses degree symbol or zero symbol to denote binary operation. However, the most commonly use is Asterisk symbol.

In binary operation, the most common operations include:

Addition of real numbers

Subtraction of real numbers

Multiplication of real numbers

Division of real numbers.

CLASS ACTIVITIES:

  1. Define binary operation
  2. List the possible operations in binary operation.

SUB TOPIC: LAWS OF BINARY OPERATIONS:

The laws of binary operations are also known as properties of binary operations.

  1. Closure Property:

Given a non-empty set is said to be closed under a binary operation if for all , . Where and are elements in (belonging to) set and means belong.

For example, the set of all integers is closed under addition, subtraction and multiplication except for division.

To illustrate non-closure of real numbers under division operation, lets consider this Example: Given, then 2 but. Hence, the is not closed under the division operation

Example 1:

Let the operation be defined on the set of real numbers by evaluate:

Solutions:

 

 

Exampled 2:

Supposed and is defined on such that for every Is closed under ?

Solution:

For every

since when two odd integers are added the result is an even number. Hence D is not closed under in other words is not closed under addition.

  1. Commutative Property:

A binary operation is said to be commutative if for each the operations addition and multiplication are commutative since and

Example: .

Also,

However, the operations ‘subtraction’ and ‘division’ are not commutative. For instance, .

Example:

Similarly, ;

  1. Associative Property:

A non- empty set which is closed under a binary operation is said to be associative if for every . The operation of addition and multiplication of real numbers are associative if is the set of real numbers and then:

(

 

Example:

  1. The operation on the set of real numbers is defined by:

, for . Determine whether or not :

  1. is commutative in ?
  2. is associative in ?

Solution:

 

Hence, the operation is commutative in

 

  1. Distributive Property:

Given a non-empty set that is closed under two binary operations and , if for all then the operation is said to be left-distributive over the operation.

Similarly, if , then the operation is said to be right- distributive over .

Example 1:

Consider ordinary multiplication and addition on the set of real numbers:

But

So, while multiplication is distributive over addition in , the set of real numbers, addition is not distributive over multiplication.

From the above example, we can generalize that; if , then :

But

Example 2:

Consider multiplication and subtraction on the set of real numbers:

 

 

From example 2 above, we can also generalize that, if then:

Hence, while multiplication distributes over subtraction, subtraction does not distribute over multiplication.

Example 3:

The operations are defined on the set of real numbers by:

for all and .

Does the operation distribute over the operation ?

Solution:

Let

 

Hence, the operation distributes over in .

CLASS ACTIVITIES:

  1. The operation on the set of rational numbers is defined by:

2 ;

Determine: (i) (ii) (iii)

  1. Verify that the operation of union of sets distributes over the operation of intersection of sets and vice visa.
  2. Given that a set of real numbers is closed under the operations and defined by:

for all

for all .

  1. Does distribute over ?
  2. Does distribute over ?

SUB TOPIC: PROPERTIES OF COMPLEMENTATION AS IN A SET

The binary operations under the complementation comprises of two elements namely; identity and inverse elements.

The identity elements:

A non-empty set that is closed under a binary operation is said to have an identity element if there exists an element that belong to the set , that is .

The identity element is also called neutral element. Following the uniqueness of identity elements, a set can not have more than one identity element under the operation if it exists.

Example 1:

An operation is defined on the set of real numbers by:

. Find the identity element.

:

let the identity element in be .

If a then,

Hence, the identity element in under is .

Example 2:

The operation on the set of real numbers is defined by for all . Find the neutral element in under the operation.

Solution:

Let the neutral element sought for be .

The inverse elements:

Given a non-empty set , that is closed under a binary opeartion , if , and an element can be found in (i.e.: Where is the neutral element in under , then is called inverse element of in .

Example 1.

The operation on the set of real number is defined by: for all find the inverse of

Solution:

We first look for the neutral element in under *. If the neutral element in under is e

Then:

 

Let the inverse of be -1

Then: -1

-1

-1

CLASS ACTIVITIES:

  1. Let be a binary operation defined on the set of real numbers by: Obtain the inverse of an element .
  2. Let be a binary operation defined on the set of real numbers by: obtain the inverse of an element

SUB TOPIC: BINARY OPERATION TABLE

The tables of the rule of binary operation for any modulo can be drawn for addition and multiplication using the appropriate rule of combination. The table helps to detect, whether or not a particular set is closed under a binary operation.

Example:

The tables below show addition and multiplication in modulo 7.

Solution:

In modulo 7, the set of numbers to be considered is: 7.

Modulo 7 addition operational Table. Modulo 7 Multiplication Operational Table

0 1 2 3 4 5 6 0 1 2 3 4 5 6
0 0 1 2 3 4 5 6 0 0 0 0 0 0 0 0
1 1 2 3 4 5 6 0 1 0 1 2 3 4 5 6
2 2 3 4 5 6 0 1 2 0 2 4 6 1 3 5
3 3 4 5 6 0 1 2 3 0 3 6 2 5 1 4
4 4 5 6 0 1 2 3 4 0 4 1 5 2 6 3
5 5 6 0 1 2 3 4 5 0 5 3 1 6 4 2
6 6 0 1 2 3 4 5 6 0 6 5 4 3 2 1

CLASS ACTIVITIES:

  1. Construct addition and multiplication operation tables for modulo 5.
  2. The operation on the set of real numbers is defined by:

for all

  1. Is the set closed under ?
  2. Is the operation commutative in ?
  3. Determine the identity element in
  4. Is the operation associative?
  5. What is the inverse of the element if ?

PRACTICE EXERCISE:

Objective Test:

  1. The operation on the set of rational numbers, is defined by: for all Find the neutral element in under
  2. 1 B. C. 2 D. E. 3

An operation is defined on the set, by Find:

  1. A. B. C. D. E.
  2. . A. B. 2 C. 3 D. 4 E. 5
  3. A. B. C. D. E.
  4. A. 5 B. 4 C. 3 D. 2 E. 1

Essay Questions

An operation is defined on the set by the following table:

a B c d
a a B c D
b b C d A
c c D a B
d d A d C
  1. Is closed under ?
  2. Is the operation commutative in
  3. What is the identity element if it exists?

An operation on the set of rational numbers is defined by: for all

  1. Is closed under
  2. Is commutative in ?

ASSIGNMENT

  1. Let the operation be defined on the set of real numbers by evaluate: (i) .
  2. Determine whether or not
  3. A binary operation * is defined on the set of R of real numbers by: (i) is the operation * closed on the set of R? Give a reason for your answer. (ii) Find the identity element of R under the operation *. (iii) determine the inverse under * of a general element aR stating which element has no inverse.
  4. A binary operation is * defined as shown in the table. Find the inverse of c, under the operation *.
* a b c d
a d c b A
b c a d b
c b d a C
d a b c d

 

  1. An operation * is defined on the set of real numbers, (a) find the identity element e of R under the operation, (b) determine the inverse of an element for which no inverse exists.

KEY WORDS

  • OPERATION
  • BINARY
  • CLOSED
  • COMMUTATIVE
  • ASSOCIATIVE
  • DISTRIBUTIVE
  • REAL NUMBERS
  • ELEMENT
  • SET
  • INVERSE
  • IDENTITY
  • RATIONAL NUMBERS