Determine whether each of the following operations define a binary ope

1.	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.
Answer» Given that ‘’ is an operation that is valid in the Natural Numbers ‘N’ and it is defined as given: ⇒ ab = 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, ⇒ ab∈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’.**

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.

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’.

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.