The algebraic operation given is: x * y = x+y-x 2.
This operation is defined on the real number set \mathbb{R}.
The nature of (1) finding * arithmetic.
We need to check the following properties:
Closure: for any x, is y \in \mathbb{R} and x * y also in \mathbb{R}?
Law of association: for any x, y, z \in \mathbb{R}, is it (x * y) * z = x * (y * z)?
Unit element: is there an element e \in \mathbb{R} that makes e * x = x * e = x for all x \in \mathbb{R}?
Inverse element: for each x \in \mathbb{R}, is there a y \in \mathbb{R} that makes x * y = y * x = e, where e is the identity element?
close
X * y = x+y-x 2 Because both X and Y are real numbers, their sum, difference and product are also real numbers, so the * operation is closed.
associative law
We need to verify that: (x * y) * z = x * (y * z) is substituted into * to calculate: (x+y-x 2) * z = x * (y+z-y 2) x+y-x 2+z-(x+y-x 2). 2 = x+y+z-y 2-x 2 Expand and simplify both sides: x+y+z-x2-y2+2xy-x2z+x2y-yz = x+y+z-y2-x2. As you can see, this is different.
unit
We need to find an e so that for all x: e * x = x * e = x e+x-e 2 = x e (1-e) = 0, that is to say, e = 0 or e = 1. We need to check which is the correct unit:
If e = 0: 0 * x = x-x 2 = x-x 2 = 0, this is true.
If e =1:1* x = x1+x-1= x, this also holds.
Therefore, both e = 0 and e = 1 can be used as unit elements.
Inverse element
For each x, we need to find a y such that: x * y = y * x = e where e is the unit. Let's consider e = 0: x+y-x 2 = 0 y = x 2-x, which means that the reciprocal of each x is y = x 2-X. 。
(2) looking for
Special elements may include:
Unit element: We have determined that e = 0 and e = 1.
Idempotent element: for x, if x * x = x, then x is an idempotent element. Substitution operation: x+x-x 2 = x 2x-x 2 = x x (2-x) = 0, which means that x = 0 or x = 2 is an idempotent.
Please note that these results are based on the given algebraic operation definition, and the actual mathematical proof may need more detailed calculation and verification.