RD Sharma Class 12 Solutions - Binary Operations

 Page 1


3. Binary Operations
Exercise 3.1
1 A. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a
b
 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a
b
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
We also know that p
q
>0 if p>0 and q>0.
So, we can state that,
? a
b
>0
? a
b
?N ...... (2)
From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.
Thus we can clearly state that ‘*’ is a Binary Operation on ‘N’.
1 B. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on Z defined by a O b = a
b
 for all a,b Z.
Answer
Given that ‘?’ is an operation that is valid in the Integers ‘Z’ and it is defined as given:
? a?b = a
b
, where a,b?Z
Since a?Z and b?Z,
According to the problem it is given that on applying the operation ‘*’ for two given integers it gives Integers
as a result of the operation,
? a?b?Z ...... (1)
Let us values of a = 2 and b = – 2 on substituting in the R.H.S side we get,
? a
b
 = 2 
– 2
? 
? a
b
?Z ...... (2)
From (2), we can see that a
b
 doesn’t give only Integers as a result. So, this cannot be stated as a binary
function.
? The operation ‘?’ does not define a binary function on Z.
1 C. Question
Determine whether each of the following operations define a binary operation on the given set or not:
Page 2


3. Binary Operations
Exercise 3.1
1 A. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a
b
 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a
b
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
We also know that p
q
>0 if p>0 and q>0.
So, we can state that,
? a
b
>0
? a
b
?N ...... (2)
From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.
Thus we can clearly state that ‘*’ is a Binary Operation on ‘N’.
1 B. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on Z defined by a O b = a
b
 for all a,b Z.
Answer
Given that ‘?’ is an operation that is valid in the Integers ‘Z’ and it is defined as given:
? a?b = a
b
, where a,b?Z
Since a?Z and b?Z,
According to the problem it is given that on applying the operation ‘*’ for two given integers it gives Integers
as a result of the operation,
? a?b?Z ...... (1)
Let us values of a = 2 and b = – 2 on substituting in the R.H.S side we get,
? a
b
 = 2 
– 2
? 
? a
b
?Z ...... (2)
From (2), we can see that a
b
 doesn’t give only Integers as a result. So, this cannot be stated as a binary
function.
? The operation ‘?’ does not define a binary function on Z.
1 C. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a + b – 2 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a + b – 2, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
Let us take the values of a = 1 and b = 1, substituting in the R.H.S side we get,
? a + b – 2 = 1 + 1 – 2
? a + b – 2 = 0?N ...... (2)
From (2), we can see that a + b – 2 doesn’t give only Natural numbers as a result. So, this cannot be stated
as a binary function.
? The operation ‘*’ does not define a binary operation on N.
1 D. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘x
6
’ on S={1,2,3,4,5} defined by a x
6
 b = Remainder when ab is divided by 6.
Answer
Given that ‘x
6
’ is an operation that is valid for the numbers in the Set S = {1,2,3,4,5} and it is defined as
given:
? ax
6
b = Remainder when ab is divided by 6, where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives one of the numbers in the set ‘S’ as a result of the operation,
? ax
6
b?S ...... (1)
Let us take the values of a = 3, b = 4,
? ab = 3×4
? ab = 12
We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given
set ‘S’.
The operation ‘x
6
’ does not define a binary operation on set S.
1 E. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘+
6
’ on S = {0,1,2,3,4,5} defined by a +
6
 b= 
Answer
Given that ‘*’ is an operation that is valid for the numbers in the set S = {0,1,2,3,4,5} and it is defined as
given:
? a +
6
 b = , where a,b?S
Page 3


3. Binary Operations
Exercise 3.1
1 A. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a
b
 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a
b
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
We also know that p
q
>0 if p>0 and q>0.
So, we can state that,
? a
b
>0
? a
b
?N ...... (2)
From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.
Thus we can clearly state that ‘*’ is a Binary Operation on ‘N’.
1 B. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on Z defined by a O b = a
b
 for all a,b Z.
Answer
Given that ‘?’ is an operation that is valid in the Integers ‘Z’ and it is defined as given:
? a?b = a
b
, where a,b?Z
Since a?Z and b?Z,
According to the problem it is given that on applying the operation ‘*’ for two given integers it gives Integers
as a result of the operation,
? a?b?Z ...... (1)
Let us values of a = 2 and b = – 2 on substituting in the R.H.S side we get,
? a
b
 = 2 
– 2
? 
? a
b
?Z ...... (2)
From (2), we can see that a
b
 doesn’t give only Integers as a result. So, this cannot be stated as a binary
function.
? The operation ‘?’ does not define a binary function on Z.
1 C. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a + b – 2 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a + b – 2, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
Let us take the values of a = 1 and b = 1, substituting in the R.H.S side we get,
? a + b – 2 = 1 + 1 – 2
? a + b – 2 = 0?N ...... (2)
From (2), we can see that a + b – 2 doesn’t give only Natural numbers as a result. So, this cannot be stated
as a binary function.
? The operation ‘*’ does not define a binary operation on N.
1 D. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘x
6
’ on S={1,2,3,4,5} defined by a x
6
 b = Remainder when ab is divided by 6.
Answer
Given that ‘x
6
’ is an operation that is valid for the numbers in the Set S = {1,2,3,4,5} and it is defined as
given:
? ax
6
b = Remainder when ab is divided by 6, where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives one of the numbers in the set ‘S’ as a result of the operation,
? ax
6
b?S ...... (1)
Let us take the values of a = 3, b = 4,
? ab = 3×4
? ab = 12
We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given
set ‘S’.
The operation ‘x
6
’ does not define a binary operation on set S.
1 E. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘+
6
’ on S = {0,1,2,3,4,5} defined by a +
6
 b= 
Answer
Given that ‘*’ is an operation that is valid for the numbers in the set S = {0,1,2,3,4,5} and it is defined as
given:
? a +
6
 b = , where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives the number that present in the set ‘S’ as a result of the operation,
? a + 
6
b?S ...... (1)
Since the addition of any two whole numbers must give a whole number, we can say that,
? a + b=0
For a + b<6, a + b?S
? a + b=6 ? a + b – 6=0?W
But according to the numbers given in the problem, the maximum value of the sum we can attain is 6.
So, we can say that a + b – 6?S.
? The operation ‘ + 
6
’ defines a binary operation on S.
1 F. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on N defined by a O b = a
b
 + b
a
 for all a,b N.
Answer
Given that ‘?’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a?b = a
b
 + b
a
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘?’ for two given natural numbers it gives
a natural number as a result of the operation,
? a?b?N ...... (1)
We know that p
q
>0 if p>0 and q>0.
? a
b
>0 and b
a
>0.
We also know that the sum of two natural numbers is a natural number.
? a
b
 + b
a
?N
? The operation ‘?’ defines the binary operation on N.
1 G. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on Q defined by  for all a,b Q.
Answer
Given that ‘*’ is an operation that is valid in the Rational Numbers ‘Q’ and it is defined as given:
? , where a,b?Q
Since a?Q and b?Q,
According to the problem it is given that on applying the operation ‘*’ for two given rational numbers it gives
a rational number as a result of the operation,
? a*b?Q ...... (1)
Let the value of b = – 1 and a = 2
Page 4


3. Binary Operations
Exercise 3.1
1 A. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a
b
 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a
b
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
We also know that p
q
>0 if p>0 and q>0.
So, we can state that,
? a
b
>0
? a
b
?N ...... (2)
From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.
Thus we can clearly state that ‘*’ is a Binary Operation on ‘N’.
1 B. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on Z defined by a O b = a
b
 for all a,b Z.
Answer
Given that ‘?’ is an operation that is valid in the Integers ‘Z’ and it is defined as given:
? a?b = a
b
, where a,b?Z
Since a?Z and b?Z,
According to the problem it is given that on applying the operation ‘*’ for two given integers it gives Integers
as a result of the operation,
? a?b?Z ...... (1)
Let us values of a = 2 and b = – 2 on substituting in the R.H.S side we get,
? a
b
 = 2 
– 2
? 
? a
b
?Z ...... (2)
From (2), we can see that a
b
 doesn’t give only Integers as a result. So, this cannot be stated as a binary
function.
? The operation ‘?’ does not define a binary function on Z.
1 C. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a + b – 2 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a + b – 2, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
Let us take the values of a = 1 and b = 1, substituting in the R.H.S side we get,
? a + b – 2 = 1 + 1 – 2
? a + b – 2 = 0?N ...... (2)
From (2), we can see that a + b – 2 doesn’t give only Natural numbers as a result. So, this cannot be stated
as a binary function.
? The operation ‘*’ does not define a binary operation on N.
1 D. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘x
6
’ on S={1,2,3,4,5} defined by a x
6
 b = Remainder when ab is divided by 6.
Answer
Given that ‘x
6
’ is an operation that is valid for the numbers in the Set S = {1,2,3,4,5} and it is defined as
given:
? ax
6
b = Remainder when ab is divided by 6, where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives one of the numbers in the set ‘S’ as a result of the operation,
? ax
6
b?S ...... (1)
Let us take the values of a = 3, b = 4,
? ab = 3×4
? ab = 12
We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given
set ‘S’.
The operation ‘x
6
’ does not define a binary operation on set S.
1 E. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘+
6
’ on S = {0,1,2,3,4,5} defined by a +
6
 b= 
Answer
Given that ‘*’ is an operation that is valid for the numbers in the set S = {0,1,2,3,4,5} and it is defined as
given:
? a +
6
 b = , where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives the number that present in the set ‘S’ as a result of the operation,
? a + 
6
b?S ...... (1)
Since the addition of any two whole numbers must give a whole number, we can say that,
? a + b=0
For a + b<6, a + b?S
? a + b=6 ? a + b – 6=0?W
But according to the numbers given in the problem, the maximum value of the sum we can attain is 6.
So, we can say that a + b – 6?S.
? The operation ‘ + 
6
’ defines a binary operation on S.
1 F. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on N defined by a O b = a
b
 + b
a
 for all a,b N.
Answer
Given that ‘?’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a?b = a
b
 + b
a
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘?’ for two given natural numbers it gives
a natural number as a result of the operation,
? a?b?N ...... (1)
We know that p
q
>0 if p>0 and q>0.
? a
b
>0 and b
a
>0.
We also know that the sum of two natural numbers is a natural number.
? a
b
 + b
a
?N
? The operation ‘?’ defines the binary operation on N.
1 G. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on Q defined by  for all a,b Q.
Answer
Given that ‘*’ is an operation that is valid in the Rational Numbers ‘Q’ and it is defined as given:
? , where a,b?Q
Since a?Q and b?Q,
According to the problem it is given that on applying the operation ‘*’ for two given rational numbers it gives
a rational number as a result of the operation,
? a*b?Q ...... (1)
Let the value of b = – 1 and a = 2
? 
? 
? 
? the operation ‘*’ does not define a binary operation on Q.
2 A. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On Z 
+
 , defined * by a*b = a – b
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Positive integers ‘Z 
+
 ’ and it is defined as given:
? a*b = a – b, where a,b?Z 
+
Since a?Z 
+
 and b?Z 
+
 ,
According to the problem it is given that on applying the operation ‘*’ for two given positive integers it gives
a positive integer as a result of the operation,
? a*b?Z 
+
 ...... (1)
Let us take the values a = 1 and b = 2
? a – b = 1 – 2
? a – b = – 1?Z 
+
? The operation * does not define a binary operation on Z 
+
2 B. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On Z 
+
 , defined * by a*b = ab
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Positive integers ‘Z 
+
 ’ and it is defined as given:
? a*b = ab, where a,b?Z 
+
 ,
Since a?Z 
+
 and b?Z 
+
 ,
According to the problem it is given that on applying the operation ‘*’ for two given positive integers it gives
a Positive integer as a result of the operation,
? a*b?Z 
+ –
 ...... (1)
We know that p q>0 if p>0 and q>0.
? ab>0?N
? ab?Z 
+
? The operation * defines a binary operation on Z 
+
.
2 C. Question
Page 5


3. Binary Operations
Exercise 3.1
1 A. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a
b
 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a
b
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
We also know that p
q
>0 if p>0 and q>0.
So, we can state that,
? a
b
>0
? a
b
?N ...... (2)
From (1) and (2) we can see that both L.H.S and R.H.S gave only Natural numbers as a result.
Thus we can clearly state that ‘*’ is a Binary Operation on ‘N’.
1 B. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on Z defined by a O b = a
b
 for all a,b Z.
Answer
Given that ‘?’ is an operation that is valid in the Integers ‘Z’ and it is defined as given:
? a?b = a
b
, where a,b?Z
Since a?Z and b?Z,
According to the problem it is given that on applying the operation ‘*’ for two given integers it gives Integers
as a result of the operation,
? a?b?Z ...... (1)
Let us values of a = 2 and b = – 2 on substituting in the R.H.S side we get,
? a
b
 = 2 
– 2
? 
? a
b
?Z ...... (2)
From (2), we can see that a
b
 doesn’t give only Integers as a result. So, this cannot be stated as a binary
function.
? The operation ‘?’ does not define a binary function on Z.
1 C. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on N defined by a * b = a + b – 2 for all a,b N.
Answer
Given that ‘*’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a*b = a + b – 2, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘*’ for two given natural numbers it gives
a natural number as a result of the operation,
? a*b?N ...... (1)
Let us take the values of a = 1 and b = 1, substituting in the R.H.S side we get,
? a + b – 2 = 1 + 1 – 2
? a + b – 2 = 0?N ...... (2)
From (2), we can see that a + b – 2 doesn’t give only Natural numbers as a result. So, this cannot be stated
as a binary function.
? The operation ‘*’ does not define a binary operation on N.
1 D. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘x
6
’ on S={1,2,3,4,5} defined by a x
6
 b = Remainder when ab is divided by 6.
Answer
Given that ‘x
6
’ is an operation that is valid for the numbers in the Set S = {1,2,3,4,5} and it is defined as
given:
? ax
6
b = Remainder when ab is divided by 6, where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives one of the numbers in the set ‘S’ as a result of the operation,
? ax
6
b?S ...... (1)
Let us take the values of a = 3, b = 4,
? ab = 3×4
? ab = 12
We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given
set ‘S’.
The operation ‘x
6
’ does not define a binary operation on set S.
1 E. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘+
6
’ on S = {0,1,2,3,4,5} defined by a +
6
 b= 
Answer
Given that ‘*’ is an operation that is valid for the numbers in the set S = {0,1,2,3,4,5} and it is defined as
given:
? a +
6
 b = , where a,b?S
Since a?S and b?S,
According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it
gives the number that present in the set ‘S’ as a result of the operation,
? a + 
6
b?S ...... (1)
Since the addition of any two whole numbers must give a whole number, we can say that,
? a + b=0
For a + b<6, a + b?S
? a + b=6 ? a + b – 6=0?W
But according to the numbers given in the problem, the maximum value of the sum we can attain is 6.
So, we can say that a + b – 6?S.
? The operation ‘ + 
6
’ defines a binary operation on S.
1 F. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘O’ on N defined by a O b = a
b
 + b
a
 for all a,b N.
Answer
Given that ‘?’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given:
? a?b = a
b
 + b
a
, where a,b?N
Since a?N and b?N,
According to the problem it is given that on applying the operation ‘?’ for two given natural numbers it gives
a natural number as a result of the operation,
? a?b?N ...... (1)
We know that p
q
>0 if p>0 and q>0.
? a
b
>0 and b
a
>0.
We also know that the sum of two natural numbers is a natural number.
? a
b
 + b
a
?N
? The operation ‘?’ defines the binary operation on N.
1 G. Question
Determine whether each of the following operations define a binary operation on the given set or not:
‘*’ on Q defined by  for all a,b Q.
Answer
Given that ‘*’ is an operation that is valid in the Rational Numbers ‘Q’ and it is defined as given:
? , where a,b?Q
Since a?Q and b?Q,
According to the problem it is given that on applying the operation ‘*’ for two given rational numbers it gives
a rational number as a result of the operation,
? a*b?Q ...... (1)
Let the value of b = – 1 and a = 2
? 
? 
? 
? the operation ‘*’ does not define a binary operation on Q.
2 A. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On Z 
+
 , defined * by a*b = a – b
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Positive integers ‘Z 
+
 ’ and it is defined as given:
? a*b = a – b, where a,b?Z 
+
Since a?Z 
+
 and b?Z 
+
 ,
According to the problem it is given that on applying the operation ‘*’ for two given positive integers it gives
a positive integer as a result of the operation,
? a*b?Z 
+
 ...... (1)
Let us take the values a = 1 and b = 2
? a – b = 1 – 2
? a – b = – 1?Z 
+
? The operation * does not define a binary operation on Z 
+
2 B. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On Z 
+
 , defined * by a*b = ab
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Positive integers ‘Z 
+
 ’ and it is defined as given:
? a*b = ab, where a,b?Z 
+
 ,
Since a?Z 
+
 and b?Z 
+
 ,
According to the problem it is given that on applying the operation ‘*’ for two given positive integers it gives
a Positive integer as a result of the operation,
? a*b?Z 
+ –
 ...... (1)
We know that p q>0 if p>0 and q>0.
? ab>0?N
? ab?Z 
+
? The operation * defines a binary operation on Z 
+
.
2 C. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On R, defined * by a*b = ab
2
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Real Numbers ‘R’ and it is defined as given:
? a*b = ab
2
, where a,b?R
Since a?R and b?R,
According to the problem it is given that on applying the operation ‘*’ for two given real numbers it gives a
real number as a result of the operation,
? a*b?R ...... (1)
We know that ab?R if a?R and b?R
? The operation * defines a binary operation on R.
2 D. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On Z 
+
 , defined * by a*b = |a – b|
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Positive integers ‘Z 
+
 ’ and it is defined as given:
? a*b = |a – b|, where a,b?Z 
+
 ,
Since a?Z 
+
 and b?Z 
+
 ,
According to the problem it is given that on applying the operation ‘*’ for two given positive integers it gives
a Positive integer as a result of the operation,
? a*b?Z ...... (1)
Let us take a = 2 and b = 2,
? |a – b| = |2 – 2|
? |a – b| = |0|
? |a – b| = 0?Z 
+
? The operation * does not define a binary function on Z 
+
 .
2 E. Question
Determine whether or not each definition * given below gives a binary operation. In the event that * is not a
binary operation give justification of this.
On Z 
+
 , defined * by a*b = a
Here, Z 
+
 denotes the set of all non – negative integers.
Answer
Given that ‘*’ is an operation that is valid in the Positive integers ‘Z 
+
 ’ and it is defined as given:
? a*b = a, where a,b?Z 
+
 ,