Question

A binary operation * is defined on the set R of real numbers by:

a*b=a+b-2ab,where a,b€R
(I)is the operation * closed on the set R?give a reason for your answer
(II)find the identity element of R under the operation*
(III)determine the inverse under * of a general element a€R stating which element has no inverse

Answer 1

[AI-generated answer — Bot GPT 3.5]

(I) To determine if the operation * is closed on the set R, we need to check if a*b is also a real number for any a, b ∈ R.

Let's take two arbitrary real numbers a and b.
a*b = a + b - 2ab

Since a, b ∈ R and real numbers are closed under addition and multiplication, a + b and 2ab are also real numbers.
Therefore, a*b is a real number for any a, b ∈ R.
Hence, the operation * is closed on the set R.

(II) To find the identity element of R under the operation *, we need to find the element e such that a * e = a for all a ∈ R.

Let e be the identity element.
a * e = a + e - 2ae = a
e = 0

Therefore, the identity element of R under the operation * is 0.

(III) To determine the inverse under * of a general element a ∈ R, we need to find an element b such that a * b = 0.

a * b = a + b - 2ab
0 = a + b - 2ab

Rearranging the equation gives us:
2ab - b = a
b(2a - 1) = a
b = a / (2a - 1)

If a = 1/2, b will be undefined since the denominator becomes zero.
Therefore, the element a = 1/2 has no inverse under the operation *.