- #1
Figaro
- 103
- 7
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.
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.
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.