1. Let S be a nonempty set and let f be the set of all functions that map S into S. Suppose that for every f and g in f we have ( f ° g)(x) = ( g ° f )(x) for all x ∈ S. Prove that S has only one...

1. Let S be a nonempty set and let f be the set of all functions that map S into S. Suppose that for every f and g in f we have

( f ° g)(x) = ( g ° f )(x) for all x ∈ S.

Prove that S has only one element

2. Mark each statement True or False. Justify each answer.

(a) Two sets S and T are equinumerous if there exists a bijection f : S → T.

(b) If a set S is finite, then S is equinumerous with In for some n ∈ N.

(c) If a cardinal number is not finite, it is said to be infinite.

(d) A set S is denumerable if there exists a bijection f : R → S.

(e) Every subset of a countable set is countable.

(f ) Every subset of a denumerable set is denumerable.