BINARY OPERATIONS Introduction You first learned of binary operations in elementary elementary (basic) school. The objects you were 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, operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, 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” operation”. By way of summary, binary operation is: Any operation or sign that combines any two elements of a given set according to some clearly defined rule. rule. 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.
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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PROPERTIES
1. Closure Property (a) Suppose the binary operation of addition 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. num ber. Under those circumstances, we say that the operation of addition a ddition “+” 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. mother. Rather, all children in the nuclear family 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 . Is the operation
Answer
by
closed on S?
From the equation
.
But
2. The binary operation
Is the operation
b
b b c
a m
c c d
b c
, therefore the operation
is defined on the set
a a
a
c d
is not closed
by the table below.
d d b
c a
c d b
closed on the set B?
Answer From the table, we notice that
, so the operation
is not closed on the set B.
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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3. The operation
is defined on the set 1
3
5
3
1
5
3
1
3
5
5
5
5
5
1
Is the operation
by the table below.
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 p roperties 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 is defined on the set R of real numbers by .
1. The binary operation
(a) Evaluate
) (ii). (i). in (i) and (ii)? (b) Find the truth set of
(iii). What conclusion can you draw from the results
, correct to two decimal places.
Answer (a) (i).
(ii).
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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. Hence, the operation
From the result of (i) and (ii), (b)
is not commutative.
Using the quadratic formula;
)
(where
.
2. A binary operation (i) Is the operation
Therefore the truth set of the equation is
.
is defined on the set R of real numbers by
commutative?
(ii) Find the truth set of the equation
.
Answer
(i)
Since Hence, the binary operation (ii)
Let
we can deduce from (1) and (2) that is commutative.
.
then the above equation becomes:
Giving
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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Thus
have no real solutions so we discard these answers.
The equation
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 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, hap pens, 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 draw from the results of (i) and (ii)? (iii). Find the truth set of of the equation
(Refer to answer on page 6)
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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Answer (i).
From the results of (i) and (ii), not associative. (iii).
(ii).
Hence the binary operation
is
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)
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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 bin ary 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)
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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 always true, hence is distributive over on the set R. (b). The equation is not true on the set R. Hence distributive over .
2. The binary operation
and
(a). Evaluate (i). from the results of (i) and (ii)?
is not
are defined on the set R of real numbers by
. and
is
(ii).
.
.
What conclusion can you draw
(b). Find the truth set of the equations Answer (a).
From equations (1) and (2), not distributive over the operation
. Hence the operation
.
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
is
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(b).
. Hence, the truth set of the equation
Thus
.
3. The binary operations
a
a
b
c
a
b
c
b c
b c
c a
and
are defined on the set S by the tables below.
a
a
(a). Determine whether or not (i). the operation commutative
Answer (a). (i). Using the table for the operation
,
c
a
c
b
b c
a b
c a
is commutative (ii). the operation
(b). Determine whether or not (i) the operation is associative (c). Determine whether or not the operation
b
b c
b
a
is associative (ii). the operation
is distributive over the operation
is
is
(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.
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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(b). (i). Using the table for the operation ,
Therefore
Also And From the equations (1), (2) and (3), we deduce that the operation is associative.
(iii). Using the table for the operation
From the equations (1) & (2), we deduce that Hence the operation
is not associative.
(c). Here we want to determine whether or not possible arrangements of a, b, c. Now and Again
Also
for all
and
and
From the equations (3), we deduce that the operation operation
is not distributive over the
. Note that, while the results from equations (1) and (2) are not conclusive, conclusive,
the result from (3) is decisive. 4. Given the binary operation
table below to compute (a). the value of (c). the value of
1 2 3 4
defined on the set
is commutative, use the
(b). the possible values of t
1
2
3
2
2x
3
2
3
4
4
1
4
1
2 3
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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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)
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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
3. The binary operation
is -6.
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 e (the identity element) must be unique, the solution means, it cannot be accepted as the identity element.
is inadmissible. It
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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Hence, the identity element of R under the binary operation
is 0.
is defined on the set M of positive numbers by Find, if it exists, the identity element of M under the binary operation
4. 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 e must be unique,
admitted as the identity element. Hence, the identity element of R under the binary operation 5. The binary operation
cannot be
is 0.
is defined on the set R of ordered pairs of real numbers by
under the binary operation
. (a). Show that (i). R is i s commutative
(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 c ommutative 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)
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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
6. Inverse Property Let e be e be the identity element under a binary operation
is (0, 1)
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 i t exists, under given binary operation is unique. unique. Notation: The inverse of the element
The inverse of b of b , topic Indices.
under a given binary operation is denoted by
is never the same as the inverse presented under the mathematical
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)
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Examples 1. The binary operation
is defined on the set R of real numbers by
.
Find (i). the identity element e of e of R under (iii). For what value of
(ii). the inverse
not defined?
Answer (i). Let e be the identity element. By definition of identity element,
where
.
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 ope ration
a
b c d
a
d a c
b
b
c
d c b a
on the set Q, where
d
a b c d
c a d b
(a). Find, giving reasons, whether or not (i). Q is closed with respect to
(iii). There is an identity element
(ii). the operation
is commutative
(b). Find, where possible, the inverse of the element a, b, c and d
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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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 p airs 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 id entity 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
(ii). Find the inverse of 6
of real numbers
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)
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(ii). Let
be the inverse of 6
SUMMARY 1. A binary operation
Any operation or sign that combines any two elements of a given set according to some clearly defined rule. rule.
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 Find
(i).
is defined on the set R of real numbers by (ii).
(iii).
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
(iv).
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3. A binary operation
is defined on the set
2
2 2
3 3
4 4
5 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
(i).
(ii).
(b). Find the truth set of the equations 5. A binary operation
?
is defined on the set R of real numbers by
4. A binary operation
(a). Find
by the table below
(i).
(iii).
(ii).
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
(a). Find
(i).
is defined on the set R of real numbers by
results of (i) and (ii)?
(ii).
.
(b). Find the truth set of the equation 7. A binary operation
. What conclusion can you draw from the
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)
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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
10. A binary operation
if
is defined on the set R of real numbers by
If
, show that
value of b for which
. Hence, find the
, giving your answer in the form
and r are rational numbers.
11. A binary operation
, p, q
is defined on the set R of real numbers by
, where
Find (a). the identity element e under 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).
13. A binary operation
(b).
is defined on the set R, of real numbers by
.
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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(a). Evaluate operation
.
(ii). the inverse
14. A binary operation
(b). Find
(iii). The inverse of
. Determine whether or not the operation
is defined on the set
is
The identity element e under e under the operation 1
2
3
by
on the table below. If denotes addition mod 6
(i). Complete and and the table below below (ii). Is the operation operation
0
(b). associative
15. A binary operation
.
is defined on the set R of real numbers by
(a). commutative
(i) the identity element e of e of R under the
4
closed on T? (iii). Find
the inverse of the element 2
.
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
a
Determine
is defined on the set
a
b
c
b
a
c
b c
a c
b c
by the table below.
c c
(a). (i). whether or not Q is closed with respect respect to
(b). whether the
operation is commutative (ii). What is the identity element e under e under the operation
(iii). Find the inverse of each elemet Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
?
20
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 identity element under the operation operation (c). Find the inverse of the set A under the operation 19. The binary operation
(i). Is
distributive over ordinary addition
20. The binary operation
is
defined on the set R of real numbers by
(iii). Is
commutative? (ii). Is
associative
of real numbers?
is defined on the set R, real numbers by
(iv).
Find (i).
(v).
21. The operation
(iii).
(vi).
is defined on the set
(a). Is the set P closed under the operation
(ii). associative
(ii).
by
?
(b). Is the operation (i). commutative
(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 a nd Q of the universal set
by
where
and
are complements of P
and Q respectively. Given that
(a). find
(i).
(b). Is the operation
(ii).
(iii).
(iv).
(i). commutative (ii). associative
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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REFERENCES Fair, J. and Bragg, S. C, Algebra 1, 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
Lecture Notes On Binary Operations, 2011 (Oteng R. Selasie, B.Ed. Mathematics)
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