A binary operation is defined on a set R, of really numbers a*b=a+b+2. Find the:

Question

A binary operation is defined on a set R, of really numbers a*b=a+b+2. Find the:

a) identity element under the operation *
b) inverse of b under the operation *?

Answer 1

If e is the identity, then

a*e = e*a = a
So, you need
a + e + 2 = a
Looks like e = -2
(Yes, -2 is really a number!)

The inverse of a means
a * a^{^-1} = the identity
So, what do you think?

Answer 2

To find the identity element under the operation , we need to find an element in the set R such that when we perform the operation with any element a, the result is equal to a itself.

Let's call the identity element e.

For any element a in R, we have:
a * e = a + e + 2

Since we want this expression to be equal to a for any a in R, it means that e + 2 should be equal to 0. Therefore, e = -2.

So, the identity element under the operation * is -2.

To find the inverse of b under the operation *, we need to find an element in the set R such that when we perform the operation * with b and the inverse element, the result is equal to the identity element.

Let's call the inverse of b as b'.

b * b' = -2

Using the definition of the operation *, we can substitute and solve for b':

b + b' + 2 = -2

Subtracting 2 from both sides:

b + b' = -4

Now, we can rearrange the equation and solve for b':

b' = -4 - b

So, the inverse of b under the operation * is -4 minus b.

Answer 3

To find the identity element under the operation , we need to find a number e such that for any number a in the set R, the equation a e = e * a = a holds true.

Let's consider an arbitrary number a in R: a * e = a + e + 2.

To find the identity element, we need to solve the equation for e. Rearranging the equation, we have:

a + e + 2 = a.

Subtracting a from both sides, we get:

e + 2 = 0.

Now, subtracting 2 from both sides, we obtain:

e = -2.

Therefore, the identity element under the operation * is -2.

To find the inverse of b under the operation *, we need to find a number x such that b * x = x * b = e, where e is the identity element we found earlier.

For b * x = x * b = e, we substitute e with -2:

b + x + 2 = 0.

Now, we solve for x:

x = -b - 2.

Thus, the inverse of b under the operation * is -b - 2.

A binary operation is defined on a set R, of really numbers a*b=a+b+2. Find the:

If e is the identity, then

a*a^-1

To find the identity element under the operation *, we need to find an element in the set R such that when we perform the operation * with any element a, the result is equal to a itself.

To find the identity element under the operation *, we need to find a number e such that for any number a in the set R, the equation a * e = e * a = a holds true.

To find the identity element under the operation , we need to find an element in the set R such that when we perform the operation with any element a, the result is equal to a itself.

To find the identity element under the operation , we need to find a number e such that for any number a in the set R, the equation a e = e * a = a holds true.