Question

How many subsets and proper subsets do the sets that contain exactly 1,2,3,4,8,10 and
20 elements have?​

Answer 1

Answer:

How many subsets and proper subsets do the sets that contain exactly 1,2,3,4,8,10 and 20 elements have?

The number of subsets in a set is the same as the cardinality of its powerset, since the powerset is the set of all of its subsets.

The cardinality of a powerset is 2 to the power of the cardinality of its set: |P(S)|=2|S| This is because for each element in the set, it could either be or not be in any given subset. Two possibilities for the first element, times two possibilities for the second element, and so on.

The number of proper subsets is one less than the number of sets, since the only “improper” subset is the set itself.

So a set with one element has 21 or two subsets, and one proper subset. Two elements means 22 or four subsets, and three proper subsets. Three elements means 23 or eight subsets, and seven proper subsets. Four elements: 24=16 subsets, and 15 proper subsets. Eight elements: 28=256 subsets, and 255 proper subsets. Ten elements: 210=1024 subsets, and 1023 proper subsets. Twenty elements: 220=1048576 subsets, and 1048575 proper subsets.

Answer 2

The number of subsets in a set is the same as the cardinality of its powerset, since the powerset is the set of all of its subsets.

The cardinality of a powerset is 2 to the power of the cardinality of its set: |P(S)|=2|S||P(S)|=2|S| This is because for each element in the set, it could either be or not be in any given subset. Two possibilities for the first element, times two possibilities for the second element, and so on.

The number of proper subsets is one less than the number of sets, since the only “improper” subset is the set itself.

So a set with one element has 2121 or two subsets, and one proper subset. Two elements means 2222 or four subsets, and three proper subsets. Three elements means 2323 or eight subsets, and seven proper subsets. Four elements: 24=1624=16 subsets, and 1515 proper subsets. Eight elements: 28=25628=256 subsets, and 255255 proper subsets. Ten elements: 210=1024210=1024 subsets, and 10231023 proper subsets. Twenty elements: 220=1048576220=1048576 subsets,