In the arithmetic of real numbers, there is a real number, namely 0, called the identity of addition because a + 0 = 0 + a = a for every real number a. This may be expressed in symbolic form by
∃z ∀a [a + z = z + a = a].
(We agree that the universe comprises all real numbers.)
(a) In conjunction with the existence of an additive identity is the existence of additive inverses. Write a quantified statement that expresses "Every real number has an additive inverse." (Do not use the minus sign anywhere in your statement.)
(b) Write a quantified statement dealing with the existence of a multiplicative identity for the arithmetic of real numbers.
(c) Write a quantified statement covering the existence of multiplicative inverses for the nonzero real numbers. (Do not use the exponent - 1 anywhere in your statement.)
(d) Do the results in parts (b) and (c) change in any way when the universe is restricted to the integers?