Problem as shown in the image For x, y R define x y to mean that x2 - 2x = y2 -

Question

Problem as shown in the image

For x, y R define x y to mean that x2 - 2x = y2 - 2y. You are given that is an equivalence relation on R. Let E be the set of all equivalence classes of R for the relation ; that is, E = { [x] | x R }. Extend addition on R to a binary operation " " on E defined by [x] [y] = [x + y]. For example [3] [4] = [3 + 4] = [7]. Display one other label for each of the equivalence classes [3] and [4]. Use the equivalence classes [3] and [4] to demonstrate that is not a well-defined operation.

Explanation / Answer

a.) [3] = [-1] since 3^2 - 2*3 = 3 and (-1)^2 - 2*(-1) = 3

[4] = [-2] since 4^2 - 2*4 = 8 and (-2)^2 - 2*(-2) = 8

b.) If "+" is well defined, then [3] + [4] = [3+4] = [7] should be equal to [-1] + [-2] = [-1 - 2] = [-3] since [3] = [-1] and [4] = [-2]. However,
[7] = 7^2 - 2*7 = 49 - 14 = 35
[-3] = (-3)^2 - 2*(-3) = 9 + 6 = 15

Therefore, [7] is not = to [-3], meaning that the "+" is not well defined.

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