1.Write an equivalent expression to ∃! x ∈ Z : P(x) (“there exists a unique x ∈ Z such that P(x)”), which is true when there is one and only one value of x in the set Z such that P(x) is true. 2....

1.Write an equivalent expression to ∃! x ∈ Z : P(x) (“there exists a unique x ∈ Z such that P(x)”), which is true when there is one and only one value of x in the set Z such that P(x) is true.

2. Write an equivalent expression to ∃∞ x ∈ Z : P(x) (“there exist infinitely many x ∈ Z such that P(x)”), which is true when there are infinitely many different values of x ∈ Z such that P(x) is true.

Here are two formulations of Goldbach’s conjecture (see Example 3.47):