Let S be the set S= {a+bk: a,bƐ R}, where k is a formal symbol. Define addition and multiplication operations on S as follows: given elements x = a+ bk and y =c+dk in S, x+y:= (a+c)+ (b+d)k, xy:= (ac+...

(a) Prove both identity laws for S. Include a short explanation (one sentence is fine)
of how you know what the identity elements are.
(b) Prove that the multiplicative inverse law is false for S. [That is, don’t just write
down a counterexample; also prove that your counterexample is valid.]

Extracted text: Let S be the set S= {a+bk: a,bƐ R}, where k is a formal symbol. Define addition and multiplication operations on S as follows: given elements x = a+ bk and y =c+dk in S, x+y:= (a+c)+ (b+d)k, xy:= (ac+ bd)+ (ad + bc)k.