How to define a addition and multiplication operators over a set R U {infinity} U {-infinity}?

Question

And, are the sets {infinity} and {-infinity} actually infinite sequences in their respective directions, such that I could define the operators as such? :

• = a(x1, ...) = (ax1, ax2, ...)
• = (x1, ...) + (y2, ...) = (x1+y1, x2+y2, ...)

Answer 1

Addition is not associative:

$$(-\infty+\infty)+\infty = (0)+\infty = \infty$$

$$-\infty+(\infty+\infty) = -\infty+(\infty) = 0$$

Answer 2

Hint: combine the first and third assumptions on the bottom line about addition and subtraction with infinities to derive a contradiction.