(Solved): part c and d please (c) a binary operation on \( \mathbb{Z} \) that is associative and has an identi ...

(c) a binary operation on \( \mathbb{Z} \) that is associative and has an identity element, but is not commutative Hint: Define \( \rho \) by cases by first "forcing" an integer to be an identity element for \( \rho \), then defining what \( a \rho b \) is when neither \( a \) nor \( b \) is the identity element. You answer to (b) may be helpful. (d) a binary operation on \( \mathbb{Z} \) that is commutative and has an identity ele ment, but is not associative. Hint: The hint in (c) may be valuable here too, except that here it i your answer to (a) that may be helpful.