Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.
a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.
b) A dyadic rational is a point x ∈ R such that x = n/2m for some n ∈ Z and m ∈ N. The set of dyadic rationals is uncountable.
c) Suppose that A and B are sets and that f : A → B is 1–1. If A is uncountable, then B is uncountable.
d) If E1, E2, . . . are finite sets, and E := E1 × E2 ×· · · := {(x1, x2, . . .) : x j ∈ E j for all j ∈ N }, then E is countable.
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