Question

(1) x + y = y + x
(2) x + (y + z) = (x + y) + z
(3) There is a unique "zero vector" such that x + 0 = x for allx
(4) For each x there is l), unique vector -x such that x + (-x) =0
(5) 1 times x equals x
(6) (c₁c₂)x =c₁(c₂x)
(7) c(x + y) = cx + cy
(8) (c₁ + c₂)x = c₁x +c₂x.

Suppose the multiplication cx is defined to produce(cx₁, 0) instead of (cx₁, cx₂).With the usual addition in R², are theeight conditions satisfied?
Please show work for each of the eight conditions.