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 ˆˆ R such that x = n/2m for some n ˆˆ Z and m ˆˆ 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

Then E is countable.

Transcribed Image Text:

E ::: El x E2 × := {(xi, x2 , . . . ) : xj E Ej for all j eN },