Question

How many elements does each of these sets have where a and b are distinct elements?

a.p({a, b,{a, b}})
b.p({∅, a,{a},{{a}}})
c.p(p(∅))?

Answer 1

a.
`\bigg|\{a,b,\{a,b\}\}\bigg|=3\implies\bigg|\mathscr P\bigg(\{a,b,\{a,b\}\bigg)\bigg|=2^3=8`

b.
`\bigg|\{\varnothing,a,\{a\},\{\{a\}\}\}\bigg|=4\implies\bigg|\mathscr P\bigg(\{\varnothing,a,\{a\},\{\{a\}\}\}\bigg)\bigg|=2^4=16`

c.
`|\varnothing|=0\implies\bigg|\mathscr P(\varnothing)\bigg|=2^0=1\implies\bigg|\mathscr P(\mathscr P(\varnothing))\bigg|=2^1=2`