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In this problem, we will explore how the cardinality of a subset S ⊆ X relates...

In this problem, we will explore how the cardinality of a subset S ⊆ X relates to the cardinality of a finite set X. (i) Explain why |S| ≤ |X| for every subset S ⊆ X when |X| = 1. (ii) Assume we know that if S ⊆ hni, then |S| ≤ n. Explain why we can show that if T ⊆ hn+ 1i, then |T| ≤ n + 1. (iii) Explain why parts (i) and (ii) imply that for every n ∈ N, every subset of hni is finite and has cardinality less than n + 1.

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