Let us define two new operations on R called ~ and ∗ by the following formulas. For real numbers a and b, let: a ~ b = a + b − 2, a ∗ b = 6a + 6b − 3ab − 10. we studied a similar operation to ~ on Z; so, take for granted that ~ is a commutative group operation on R. (a) Prove that (R, ~, ∗) is a commutative ring with identity (be sure to identify the additive identity and the multiplicative identity; I suggest proving that ∗ is commutative before proving the distributive laws). (b) Determine the units in (R, ~, ∗) and find their (multiplicative) inverses.

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