Question

g. Exactly 14 proper subsets h. Exactly 15 proper subsets How many elements does A contain if it h a. 64 subsets? b. 31 proper subsets? c. No proper subset? d. 255 proper subsets?

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Answer 1

I don't know what you mean by g. and h., so I'll just skip that part.

I'm assuming you're asking about some arbitrary finite set $A$.

a. If $A$ has 64 subsets, then $A$ has $\log_2(64) = \boxed{6}$ elements. This is because a set of $n$ elements has $2^n$ subsets/elements in its power set.

b. A proper subset is a subset that doesn't contain all the elements of the parent set. This means we exclude the set $A$ from its power set. The power set itself would have 32 elements, so $A$ would have $\log_2(32) = \boxed{5}$ elements.

c. The empty set is a proper subset of any non-empty set. However, if $A=\emptyset$, then it has no proper subsets. So $A$ must be the empty set and have $\boxed{0}$ elements.

d. By the same reasoning as in part (b), if $A$ has 255 proper subsets, then it has a total of 256 subsets, and $\log_2(256) = \boxed{8}$.