1. The problem statement, all variables and given/known data Let V be the set of all ordered pairs of real numbers. Suppose we define addition and scalar multiplication of elements of V in an unusual way so that when u=(x1, y1), v=(x2, y2) and k∈ℝ u+v= (x1⋅x2, y1+y2) and k⋅u=(x1/k, y1/k) Show detailed calculations of one case where V i) satisfies one of the addition axioms (1 – 5) ii) fails to uphold one of the addition axioms (1 – 5) iii) satisfies one of the scalar multiplication axioms (6 – 10) iv) fails to uphold one of the scalar multiplication axioms (6 – 10) 2. Relevant equations 3. The attempt at a solution i)u+v=v+u: Axiom 2 u+v= (x1⋅x2, y1+y2) v+u= (y1⋅y2, x1+x2) ∴u+v≠v+u ii) u+(-u)= (-u)+u=(0,0): axiom 4 (x1, y1)+(-x1,-y1)= (x1⋅-x1, y1+-y1) = (-x1, 0) iii) 1u=u: axiom 10 ku= (x1/k, y1/k) if k=1 1u= (x1/1, y1/1)= (x1,y1)=u iv)?? I don't know if I am even on the right track with the three that I have calculated. I feel confident in the third but the others, not so much. I also have struggled to demonstrate the fourth. Any help would be greatly appreciated.