Let A and B be two nonempty sets. If there is an injection of B into A, but no injection o...

Let A and B be two nonempty sets. If there is an injection of B into A, but no injection of A into B, we say that A has greater cardinality than B

(a) Conclude from Theorem 9.1 that every uncountable set has greater cardinality than .

(b) Show that if A has greater cardinality than B, and B has greater cardinality than C, then A has greater cardinality than C.

(c) Find a sequence A1, A2, … of infinite sets, such that for each n ∈ , the set An+1 has greater cardinality than An.

(d) Find a set that for every n has cardinality greater than An.