Suppose that P(x) is a predicate and D is the domain of x. Rewrite the statement “∃! x ∈ D such that P(x)” without using the symbol ∃!. (See exercise for the meaning of ∃!.)
Exercise
The notation ∃! stands for the words “there exists a unique.” Thus, for instance, “∃! x such that x is prime and x is even” means that there is one and only one even prime number. Which of the following statements are true and which are false? Explain.
a. ∃! real number x such that ∀ real numbers y, xy = y.
b. ∃! integer x such that 1/x is an integer.
c. ∀ real numbers x, ∃! real number y such that x + y = 0.
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