the operation defined by a*b=square root of ab is on the set of real numbers R.

okay, if you say so

that is the geometric mean of a and b

Ah, the mathematical world, where operations are defined with the wave of a wand, or in this case, the square root of ab. Now you've got me interested in this * operation on the set of real numbers, R. Let's take a closer look, shall we?

First, let's start with addition. I guess we can have a little fun with it. If we take two real numbers a and b, we would have a + b. But since we're using the * operation instead, we'll have to tweak it a bit. So, let's see... a * b would be... (a * b)^2 = √(ab).

Now, let's test this operation with a couple of real numbers. Let's try a = 3 and b = 4. So, according to our operation, 3 * 4 would be (3 * 4)^2 = √(3 * 4).

The question is, what number times itself gives us 12? Well, I guess the square root of 12 is around 3.464. So, unless we're working with approximations, it seems our * operation isn't exactly in line with the usual multiplication.

In conclusion, the operation * defined as √(ab) might not necessarily align with our usual expectations of multiplication. So, let's just say it adds a little twist to the mathematical world, like a clown at a math party!

To show that the operation defined by a*b = √(ab) is a binary operation on the set of real numbers ℝ, we need to verify the following properties:

1. Closure: For any two real numbers a and b, the result of a*b is also a real number.
2. Associativity: The operation is associative if for any three real numbers a, b, and c, the expression (a*b)*c is equal to a*(b*c).
3. Identity Element: There exists an identity element in the set, such that for any real number a, the expression a*identity = identity*a = a.
4. Inverse Element: For any real number a, there exists an inverse element such that a*inverse = inverse*a = identity.

Let's now check each of these properties step-by-step:

1. Closure:
To show closure, we need to check if the result of a*b is a real number for any two real numbers a and b.

Let's consider a = 4 and b = 9.
a*b = √(4 * 9) = √36 = 6, which is a real number. So closure is satisfied.

2. Associativity:
To show associativity, we need to check if (a*b)*c = a*(b*c) for any three real numbers a, b, and c.

Let's consider a = 2, b = 3, and c = 5.
(a*b)*c = (√(2 * 3)) * 5 = (√6) * 5 = 5√6

a*(b*c) = 2 * (√(3 * 5)) = 2 * (√15) = 2√15

We can see that (a*b)*c = a*(b*c), so associativity is satisfied.

3. Identity Element:
To find the identity element, we need to find a real number e such that for any real number a, a*e = e*a = a.

Let's consider a = 7.
a*e = 7 * √e = 7√e
e*a = √e * 7 = 7√e

For this to be true, we need 7√e to be equal to 7 for any value of a. This is only true when √e = 1, which implies e must be equal to 1.

So, the identity element is e = 1.

4. Inverse Element:
To find the inverse element, for any real number a, we need to find a real number b such that a*b = b*a = e (the identity element).

Let's consider a = 3. We need to find b such that 3*b = b*3 = 1.

3*b = 1
b = 1/3

So, the inverse of 3 is 1/3.

From the above analysis, we can conclude that the operation defined by a*b = √(ab) is a binary operation on the set of real numbers ℝ.

To determine if the operation defined by a*b = square root of (ab) is well-defined on the set of real numbers (R), we need to check whether it satisfies two properties:

1. Closure: The operation should produce a real number when applied to any two real numbers.
2. Associativity: The operation should be associative, meaning that (a*b)*c = a*(b*c) for any real numbers a, b, and c.

Let's now check these properties:

1. Closure: For any two real numbers a and b, the product ab will always be a real number since the real numbers are closed under multiplication. Now, we need to check if the square root of ab also yields a real number. This is true because the square root of a non-negative real number is always a real number. So, the operation defined by a*b = square root of (ab) is closed on the set of real numbers.

2. Associativity: To check associativity, we need to verify whether (a*b)*c = a*(b*c) for all real numbers a, b, and c. Let's evaluate both sides of the equation:

(a*b)*c = (sqrt(ab))*c = sqrt((sqrt(ab))*c) = sqrt(sqrt(ab)*c)
a*(b*c) = a*(sqrt(bc)) = sqrt(a*(sqrt(bc))) = sqrt(sqrt(ab*c))

It is not immediately obvious that these two expressions are equal since there are square roots involved. Therefore, we will choose specific values for a, b, and c to verify associativity.

Let's say a = 2, b = 3, and c = 4. Calculating both sides:

(2*3)*4 = (sqrt(2*3))*4 = sqrt(6)*4 ≈ 4.89898
2*(3*4) = 2*(sqrt(3*4)) = sqrt(12) ≈ 3.46410

Since (2*3)*4 is not equal to 2*(3*4) for the chosen values of a, b, and c, the operation defined by a*b = square root of (ab) is not associative.

In conclusion, the operation defined by a*b = square root of (ab) is closed on the set of real numbers (R) but it is not associative.