Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 1

BINARY OPERATIONS

Introduction

You first learned of binary operations in elementary (basic) school. The objects you were

using were mainly numbers and the binary operations you investigated were addition,

subtraction, multiplication, and division.

The concept of binary operations in mathematics has been defined by many based on

their different understanding of the concept.

Few are:

- A binary operation is a calculation involving two operands, in other words, an

operation whose arity is two. Examples include the familiar arithmetic operations

of addition, subtraction, multiplication and division.

- A binary operation is simply a rule for combining two objects of a given type, to

obtain another object of that type.

Considering all the above definitions, I can confidently summarise the idea of a binary

operation as; “a way to produce an element of a set from a given pair of elements of the

same set according to a rule of operation”.

By way of summary, binary operation is:

In studying binary operations on sets, we tend to be interested in those operations that

have certain properties and we will discuss this further.

Let us begin by taking a closer look at the properties of binary operations.

These properties form the basis for a better understanding of the concept of binary

operations.

Remember that, the properties of binary operation are in application in many sectors or

industries of our economy including the stock market, transportation, IT, energy,

medicine, printing, etc.

Any operation or sign that combines any two elements of a

given set according to some clearly defined rule.

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 2

PROPERTIES

1. Closure Property

(a) Suppose the binary operation of addition “+” is defined on a set of real numbers.

We notice that whenever two real numbers are added, we still obtain a real number.

Under those circumstances, we say that the operation of addition “+” is closed on the

set of real numbers.

(b) Suppose we define the binary operation of addition “+” on the set , we

notice that 3+7=10, but so “+” is not closed on the set A.

In general, the binary operation defined on a set S is closed on S if and only if

This resembles the idea of a nuclear family where there is no “step” son or daughter by

either the father or mother. Rather, all children in the nuclear family come from the

same source [same father, same mother]. In that sense, the set of nuclear family is closed.

No intruders or strangers.!!

Examples

1. The binary operation is defined on the set by

. Is the operation closed on S?

Answer

From the equation

. But , therefore the operation is not closed

2. The binary operation is defined on the set by the table below.

Is the operation closed on the set B?

Answer

From the table, we notice that , so the operation is not closed on the set B.

a b c d a a b c d

b c d b c

c a b c d d m c a b

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 3

3. The operation is defined on the set by the table below.

Is the operation closed on S?

Answer

Since all the answers in the table (numbers in the shaded region) are members of set S,

the operation is closed on the set S.

2. Commutative Property

The binary operation defined on the set S is said to be commutative if and only if

This property is properly one of the oldest and most frequently used properties that we

as students of mathematics have been using since our elementary days. Changing of

positions and still getting the same answer. Remember? .

Therefore, the commutative property is a very simple one that does not need much

explanation.

Examples

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

.

(a) Evaluate (i). ) (ii). (iii). What conclusion can you draw from the results

in (i) and (ii)?

(b) Find the truth set of , correct to two decimal places.

Answer

(a) (i).

(ii).

1 3 5

1 3 1 5

3 1 3 5

5 5 5 5

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 4

From the result of (i) and (ii), . Hence, the operation is not commutative.

(b)

Using the quadratic formula;

(where )

. Therefore the truth set of the equation is .

2. A binary operation is defined on the set R of real numbers by

(i) Is the operation commutative?

(ii) Find the truth set of the equation .

Answer

(i)

Since we can deduce from (1) and (2) that .

Hence, the binary operation is commutative.

(ii)

Let then the above equation becomes:

Giving

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 5

Thus

The equation

have no real solutions so we discard these answers.

The solution of the equation

Hence the truth set of the equation

3. Associative Property

If the binary operation is closed on the set S such that;

Then the operation is said to be associative.

Associative property is also one of the frequently used properties in mathematics.

At the elementary level, we have probably heard of BODMAS – Bracket of Division,

Multiplication, Addition and Subtraction.

Meaning, anytime you are working with numbers, you consider those in brackets first. In

other words, the operations in the bracket take precedence over all other operations.

In plain language, no matter what happens, you work the bracket first.

Hence, associative property says, if you work what is in the brackets first, change the

position of the brackets and do the operation the other way and the answer must be the

same.

*(Using the concept of mixing Gari, Sugar and Milo in preparation of “soakings” to explain

the concept further, we can come to the conclusion that, the resultant product is the

same irrespective of the process of combination).

Examples

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

Find (i). (ii).

What conclusion can you draw from the results of (i) and (ii)? (iii). Find the truth set of

the equation

(Refer to answer on page 6)

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 6

Answer

(i).

(ii).

From the results of (i) and (ii), Hence the binary operation is

not associative.

(iii).

.

Therefore the truth set of the equation is

.

2. The binary operation is defined on the set R of real numbers by

.

(a). Is the operation associative?

(b). Find the truth set of the equation correct to 2 decimal places.

Answer

(a). Let (since associative property deals with three variables)

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 7

R.H.S L.H.S

From the above equations, . Hence, the operation is associative.

(b).

Using the factor theorem, let

. is a factor of .

The other factors are determined by means of long division method as shown below:

Thus (Factorizing)

(Solving the equation using the quadratic equation or the general formula)

.

Hence the truth set of the equation

4. Distributive Property

The distributive property uses two binary operations on the same set. If the binary

operation and are defined on the set S such that;

Then the operation is distributive over the operation .

The Distributive property is simple. It is just a matter of one operation having authority

over another operation.

If an operation overrules another operation, we say the overruling operation is

distributive over the other.

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 8

Examples

1. The binary operations of addition and multiplication are defined on the set R

of real numbers. Determine whether or not

(a). is distributive over (b). is distributive over

Answer

(a). Let a, b, c be any real numbers, then the equation is

always true, hence is distributive over on the set R.

(b). The equation is not true on the set R. Hence is not

distributive over .

2. The binary operation and are defined on the set R of real numbers by

. and .

(a). Evaluate (i). (ii). . What conclusion can you draw

from the results of (i) and (ii)?

(b). Find the truth set of the equations

Answer

(a).

From equations (1) and (2), . Hence the operation is

not distributive over the operation .

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 9

(b).

Thus . Hence, the truth set of the equation

.

3. The binary operations and are defined on the set S by the tables below.

(a). Determine whether or not (i). the operation is commutative (ii). the operation is

commutative

(b). Determine whether or not (i) the operation is is associative (ii). the operation is

associative

(c). Determine whether or not the operation is distributive over the operation

Answer

(a). (i). Using the table for the operation , (ii) Using the table for the operation ,

Hence, the operation is not commutative

From the equations (1), (2), and (3), we deduce

that the operation is commutative.

a b c

a a b c

b b c a

c c a b

a b c

a a c b

b b a c

c c b a

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 10

(b). (i). Using the table for the operation , (iii). Using the table for the operation

Therefore From the equations (1) & (2), we deduce that

Also Hence the operation is not associative.

And

From the equations (1), (2) and (3), we deduce

that the operation is associative.

(c). Here we want to determine whether or not for all

possible arrangements of a, b, c.

Now and

Again and

Also and

From the equations (3), we deduce that the operation is not distributive over the

operation . Note that, while the results from equations (1) and (2) are not conclusive,

the result from (3) is decisive.

4. Given the binary operation defined on the set is commutative, use the

table below to compute

(a). the value of (b). the possible values of t

(c). the value of

1 2 3 4

1 2 2x 3

2 2 3 1

3 4 1 2

4 4 3

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 11

Answer

Using the table for the operation,

(Since is commutative)

Also Again

Hence Finally,

q

Solving the equations (1) and (2) simultaneously gives

Now

(a). (b). The possible values of t are -3 and 1

(c).

5. Identity Property

The set P, is said to have an identity element under a given operation if and only if

The identity element, if it exists, is unique. That is to say, the set P has only one identity

element.

From the basic education level, we have been working with identity elements.

Consider the operation of addition and multiplication .

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 12

We remember that, the naturally occurring identity of addition is 0. Whiles that of

multiplication is 1. [Recall and ].

It is simply the neutral number that when combined with any other element of a set

under a given operation leaves the value of the element unaltered.

Examples

1. Find the identity element of the set Z of positive integers under the binary operation

of addition .

Answer

Let be the identity element of Z under the operation .

Then by definition,

. Since , the set Z has no identity element under the operation .

You must remember that some sets may not have identity element under a given binary

operation.

2. The binary operation is defined on the set R of real numbers by

. Find the identity element of R under the operation . Answer

Let e be the identity element of R under the operation . Then by definition,

. Hence the identity element of R under the binary operation is -6.

3. The binary operation is defined on the set R of real numbers by

Find the identity element of R under the binary

operation .

Answer

Let e be the identity element, then by definition, Thus

.

Since e (the identity element) must be unique, the solution

is inadmissible. It

means, it cannot be accepted as the identity element.

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 13

Hence, the identity element of R under the binary operation is 0.

4. The binary operation is defined on the set M of positive numbers by

Find, if it exists, the identity element of M under the

binary operation Answer

Let e be the identity of M under the binary operation Then (By definition)

Thus

Since could be any member of the set M, and e must be unique,

cannot be

admitted as the identity element.

Hence, the identity element of R under the binary operation is 0.

5. The binary operation is defined on the set R of ordered pairs of real numbers by

. (a). Show that (i). R is commutative

under the binary operation (ii). R is associative under the binary operation .

(b). Find the identity element of R under the binary operation .

Answer

(a). But [Note that

Hence, the set R is commutative under the binary operation

(b). Let then we wish to show whether or not

Now, From the equations (1) and (2), we can conclude that

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 14

Hence the binary operation is associative on the set R. (c). Let be the identity element of R under the binary operation

Then by definition,

Hence, the identity element of R under the binary operation is (0, 1)

6. Inverse Property

Let e be the identity element under a binary operation defined on the set S,

, then the element , is said to be the inverse of if and only if

The inverse of an element, if it exists, under given binary operation is unique.

Notation:

The inverse of the element under a given binary operation is denoted by

The inverse of b, is never the same as the inverse presented under the mathematical

topic Indices.

Inverse is simply the opposite of something.

For example, the opposite of positive 2 under the operation of addition is negative 2

i.e. (-2). Implying that, if you add the two values, the resulting number is zero (0); which is

the identity element of addition .

In addition, the opposite of positive 2 under the operation of multiplication is

(from

Indices). Implying that, if you multiply the two values, the resulting value is one (1); which

is the identity element of multiplication

.

This implies that, the inverse element of a number (in a given set) is the number that

makes results in the identity element of the given operation when operated on.

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 15

Examples

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

.

Find (i). the identity element e of R under (ii). the inverse where

(iii). For what value of not defined?

Answer

(i). Let e be the identity element. By definition of identity element, .

Hence

(ii). Also by definition of inverse, .

(iii). is defined when .

Thus, the value of for which is not defined is when

2. The table below defines the binary operation on the set Q, where

(a). Find, giving reasons, whether or not

(i). Q is closed with respect to (ii). the operation is commutative

(iii). There is an identity element

(b). Find, where possible, the inverse of the element a, b, c and d

a b c d

a b d a c

b d c b a

c a b c d

d c a d b

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 16

Answers

(a). (i). Q is closed with respect to the operation since all possible answers (in the table

given above) are members of Q. [Refer to the Closure property for details]

(ii). from the table given above,

From the above equations, we see that all possible pairs of elements are commutative

with respect to . Hence the binary operation is commutative.

(iii). From the table given above,

From equations (1), (2), (3) and (4), the identity element is c.

Because by observation, any element operated on c results in the same element.

(b). from the table given above,

Since c is the identity element under the binary operation in the table above, the

inverse element of a, = d , b = b, c = c and d is c. [By definition of Inverse].

3. The binary operation is defined on the set of R of real numbers by

. (i). Find the inverse of t under the binary operation of real numbers

(ii). Find the inverse of 6

Answer

(i). Let be the inverse element of t and e be the identity element

By the definition of an inverse,

But

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 17

(ii). Let be the inverse of 6

SUMMARY

1. A binary operation

2. Closure Property

3. Commutative Property

4. Associative Property

5. Distributive Property

6. Identity Property

7. Inverse Property

SUPPLEMENTARY QUESTIONS

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

where Find (i). (ii). (iii).

(iv).

2. The binary operation is defined on the set R of real numbers by

Find (i). (ii). (iii). (iv).

Any operation or sign that combines any two elements of a

given set according to some clearly defined rule.

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 18

3. A binary operation is defined on the set by the table below

2 3 4 5

2 2 3 4 5

3 3 4 5 2

4 4 5 1 3

5 5 1 3 4

(a). Determine whether or not

(i) Q is closed with respect to the binary operation

(ii). the operation is commutative

(b). What is the identity element, e, under the operation ?

(c). Find the inverse of each element

4. A binary operation is defined on the set R of real numbers by

(a). Find (i). (ii). (iii).

(b). Find the truth set of the equations (i). (ii).

5. A binary operation is defined on the set R of real numbers by

. (a). Find (i). (ii). (iii).

(b). Find the truth set of the equation .

6. A binary operation is defined on the set R of real numbers by

.

(a). Find (i). (ii). . What conclusion can you draw from the

results of (i) and (ii)?

(b). Find the truth set of the equation

7. A binary operation is defined on the set R of real numbers by

(a). Find (i). (ii).

(b). Find the truth set of the operation (i). (ii).

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 19

8. The binary operations and are defined on the set R of real numbers by

(a). Find the value of (i). (ii).

What conclusions can you draw from the result of (i) and (ii)?

9. A binary operation is defined on the set by

.

(a). Find (i). (ii). What conclusion can you draw from

the results of (i) and (ii)?

(b). Find the values of and if

10. A binary operation is defined on the set R of real numbers by

If , show that

. Hence, find the

value of b for which , giving your answer in the form , p, q

and r are rational numbers.

11. A binary operation is defined on the set R of real numbers by

, where

Find (a). the identity element e under the operation (b). the inverse of an element ,

stating the value for which no inverse exists.

12. A binary operation is defined on the set R of real numbers

, where .

Find (a). (b).

13. A binary operation is defined on the set R, of real numbers by

.

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 20

(a). Evaluate (b). Find (i) the identity element e of R under the

operation . (ii). the inverse (iii). The inverse of .

14. A binary operation is defined on the set R of real numbers by

. Determine whether or not the operation is

(a). commutative (b). associative

15. A binary operation is defined on the set by

on the table below. If denotes addition mod 6

(i). Complete and the table below (ii). Is the operation closed on T? (iii). Find .

The identity element e under the operation the inverse of the element 2

0 1 2 3 4 5

0

1

2

3

4

5

16. A binary operation is defined on the set R of real numbers by .

Find, under the operation , the (a). the identity element (b). inverse of 1

(c). inverse of 2

17. A binary operation is defined on the set by the table below.

a b c

a b a c

b a b c

c c c c

Determine (a). (i). whether or not Q is closed with respect to (b). whether the

operation is commutative (ii). What is the identity element e under the operation ?

(iii). Find the inverse of each elemet

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 21

18. Suppose A and B are any subsets of the universal set U. The binary operation

defined on U is given by denotes the

complement of A and B respectively.

(a). Is A commutative? (b). Show that the identity element under the operation is

(c). Find the inverse of the set A under the operation

19. The binary operation defined on the set R of real numbers by

(i). Is commutative? (ii). Is associative

(iii). Is distributive over ordinary addition of real numbers?

20. The binary operation is defined on the set R, real numbers by

Find (i). (ii). (iii).

(iv). (v). (vi).

21. The operation is defined on the set by

(a). Is the set P closed under the operation ? (b). Is the operation (i). commutative

(ii). associative (c). Find e, the identity element under the operation (d). Find the

inverse of each element

22. A binary operation is defined on the subsets P and Q of the universal set

by where and are complements of P

and Q respectively. Given that

(a). find (i). (ii). (iii). (iv).

(b). Is the operation (i). commutative (ii). associative

Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics) 22

REFERENCES

Fair, J. and Bragg, S. C, Algebra 1, New Jersey: Prentice-Hall Inc., 1993

L. Bostock and S. Chandler, Mathematics-The Core Course for Advanced Level ,

Cheltenham: Stanley Thornes, 1986

WEBSITES

www.wikipedia.org

www.algebralab.com

www.math.csusb.edu

www.mathworld.wolfram.com