Question

Q1.6. Consider the binary operation on the set of real numbers, defined real
numbers defined by a*b =ab÷4

a) Prove that is commutative and associative
b) Find the identity element of on R
c) Find the inverse of 5.​

Answer 1

Step-by-step explanation:

SOLUTION:

TO CHECK:

Consider the binary operation on the set of real numbers, defined real numbers defined by

a*b =ab/4

a) Prove that is commutative and associative

b) Find the identity element of on R

c) Find the inverse of 5.

ANSWER TO QUESTION : 1

CHECKING FOR COMMUTATIVE

Let a , b ∈ R

Now

a*b =ab/4

b*a = ba/4 = ab/4

∴ a*b = b*a

So * is commutative

CHECKING FOR ASSOCIATIVE

Let a , b , c ∈ R

Now

a*(b*c) =a*(bc/4) = abc/16

(a*b)*c = (ab/4)*c = abc/16

∴ a*(b*c) = (a*b)*c

So * is associative

ANSWER TO QUESTION : 2

Let e ∈ R be the identity element

Then a*e = e*a = a

Now a*e = a gives

ae/4 = a

∴ e = 4

So 4 is the identity element

ANSWER TO QUESTION : 3

Let a be the inverse of 5

Then

a*5 = e

⇒ 5a/4 = 4

⇒ a = 16/5

So 16/5 is the inverse of 5

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1.1. Prove that the inverse of the product of two elements of a group is the product of the inverses taken in the reverse order

https://brainly.in/question/22739109

2. Show that the set Q+ of all positive rational numbers forms an abelian group under the operation * defined by

a*b= 1/2(...

https://brainly.in/question/21467351