On the set Z of all integers a binary operation * is defined by a * b

1.	On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.
Answer» *The given binary operation is ab = a+b+2** In order to find the inverse of the relation, we have to find the identity element first. Let that identity element be e then ae = a From que. ae = a+e+2 So, from the above two relations we have a+e+2 = a or, e+2 = 0 ∴ e = -2 Hence the identity element is -2 for this binary operation. Now let a’ be the inverse of this relation Then as per the definition of the inverse element aa’ = e ∴ a+a’+2 = -2 ∴ a’ = -4 -a And for 4 ,i.e. a = 4 a’ = -4 – 4 ∴ a’ = -8 Thus the inverse element of 4 is -8 for the given binary operation.*

On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.

Answer»

The given binary operation is a*b = a+b+2

In order to find the inverse of the relation, we have to find the identity element first.

Let that identity element be e then

a*e = a

From que.

a*e = a+e+2

So, from the above two relations we have

a+e+2 = a

or, e+2 = 0

∴ e = -2

Hence the identity element is -2 for this binary operation.

Now let a’ be the inverse of this relation

Then as per the definition of the inverse element

a*a’ = e

∴ a+a’+2 = -2

∴ a’ = -4 -a

And for 4 ,i.e. a = 4

a’ = -4 – 4

∴ a’ = -8

Thus the inverse element of 4 is -8 for the given binary operation.

Discussion

No Comment Found

Related InterviewSolutions

Determine whether each of the following operations define a binary operation on the given set or not: ‘*’ on N defined by a * b = ab for all a,b N.
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 = ab for all a,b Z.
Determine whether each of the following operations define a binary operation on the given set or not: ‘x6’ on S={1,2,3,4,5} defined by a x6 b = Remainder when ab is divided by 6.
On R – {1}, a binary operation * is defined by a * b = a + b – ab. Prove that * is commutative and associative. Find the identity element for * on R – {1}. Also, prove that every element of R – {1} is invertible.
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.
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.
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 = ab2Here, Z+ denotes the set of all non – negative integers.
Let * be a binary operation on the set Q of all rational numbers given as a * b = (2a – b)2 for all a, b ∈ Q. Find 3 * 5 and 5 * 3. Is 3 * 5 = 5 * 3?
Let S be the set of all real numbers except – 1 and let ‘*’ be an operation defined by a*b = a + b + ab for all ab∈S. Determine whether ‘*’ is a binary operation on ‘S’. if yes, Check its commutativity and associativity. Also, solve the equation (2*x)*3 = 7.
The binary operation * defined on R×R → R is defined as a*b = 2a + b. Find (2*3)*4.