In each of the following cases explain what is meant by the statement and decide whether it is true or false.
(i) For each x ∈ Z there exists y ∈ Z such that x + y =1.
(ii) There exists y ∈ Z such that for each x ∈ Z, x + y =1.
(iii) For each x ∈ Z there exists y ∈ Z such that xy = x.
(iv) There exists y ∈ Z such that for each x ∈ Z, xy = x.
By now you might have guessed that a for all statement can be rewritten as an if then statement. For example,
the statement for all m ∈ N, m ∈ Z is equivalent to if m ∈ N then m ∈ Z.