Question

Given the set s = { ∅, d, { d }, { { d } }, { { { d } } } }. determine the cardinality of the power set of s list the elements of the power set. to save writing, let v = ∅, w = d, x = { d }, y = { { d } }, and z = { { { d } } }.

Answer 1

`S=\{\varnothing,d,\{d\},\{\{d\}\},\{\{\{d\}\}\}\}=\{v,w,x,y,z\}`

A set containing
`n` elements has a power set containing
`2^n` elements; here,
`|S|=5` so
`|\mathscr P(S)|=2^5=32`.

The elements of the power set are all possible combinations of up to 5 of the total 5 elements to choose from:

0 choices (1):
`v` (since
`v=\varnothing`)

1choice (5):
`\{v\},\{w\},\{x\},\{y\},\{z\}`

2 choices (10):
`\{v,w\},\{v,x\},\{v,y\},\{v,z\},\{w,x\},\{w,y\},\{w,z\},\{x,y\},\{x,z\},\{y,z\}`

3 choices (10):
`\{v,w,x\},\{v,w,y\},\{v,w,z\}` and so on

4 choices (5);
`\{v,w,x,y\},\{v,w,x,z\}` and so on

5 choices (1):
`\{v,w,x,y,z\}`

(1 + 5 + 10 + 10 + 5 + 1 = 32)