(i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S) uncountable? Justify your answer.

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Plz solve d part( ii and iii) part only
(d) Let S = {2, 3, 5, 7,...} be the set of all prime numbers, and let P(S) = {T |TC S} be the
power set of S.
(i) Show that P(S) is uncountable.
Next, let F(S) (T |TC S and T is finite) be the set of all finite subsets of S, and define the
function f: F(S) → N by setting f({p-P2...Pk}) = PIP2 . Pk for each non-empty set of
k primes, (k e N), as well as f(Ø) = 1.
(i) Show that F(S) is an infinite set, and that f is injective but not surjective.
(i) Is F(S) uncountable? Justify your answer.
Transcribed Image Text:(d) Let S = {2, 3, 5, 7,...} be the set of all prime numbers, and let P(S) = {T |TC S} be the power set of S. (i) Show that P(S) is uncountable. Next, let F(S) (T |TC S and T is finite) be the set of all finite subsets of S, and define the function f: F(S) → N by setting f({p-P2...Pk}) = PIP2 . Pk for each non-empty set of k primes, (k e N), as well as f(Ø) = 1. (i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S) uncountable? Justify your answer.
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