Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets.(adsbygoogle = window.adsbygoogle || []).push({});

a) {1}

Number of Subsets: ##2^1 = 2## (∅, {1}) where the power means how many elements

Number of Possible Relations (suppose those two subsets A and B are arbitrary): ##2^2 = 4## (A⊆A, A⊆B, B⊆B, B⊆A) where the base is 2 since there are two subsets and the power is 2 since there are two possible ways to relate each pair.

Number of "True" Possible Relations: 3 (∅⊆∅, ∅⊆{1}, {1}⊆{1})

b) {1,2}

Number of Subsets: ##2^2 = 4##

Number of Possible Relations (suppose those two subsets A and B are arbitrary): ##4^2 = 16##

Number of "True" Possible Relations: 9

c) {1,2,3}

Number of Subsets: ##2^3 = 8##

Number of Possible Relations (suppose those two subsets A and B are arbitrary): ##8^2 = 64##

Number of "True" Possible Relations: 27

d) {1,2,3, ..., n}

Number of Subsets: ##2^n##

Number of Possible Relations (suppose those two subsets A and B are arbitrary): ##(2^n)^2 = 2^{2n}##

Number of "True" Possible Relations: ##3^n##

Part d) is guesswork. So is part d) correct by the pattern that is given in part a)-c)? I know that I need to do induction in order to formally say that IT is correct but the question is if I can guess by the pattern.

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# I Sets, Subsets, Possible Relations

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