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(∅))

a) P({a, b, {a, b}})=

b) P({∅, a, {a}, {{a}}})=

c) P(P(∅))=P(P{∅})=2

Step-by-step explanation:

Previous concepts

For this case we need to remember the concpts of subset and the power set.

A is a subset of B if every element of A is an eelement if B and we write

The power set of a set R is th set of all the possible subsets of R and we writ this as P(R)

Solution to the problem

Part a

P({a, b, {a, b}}) as we can see this set contains 3 elements a,b and {a,b}

And since the power set contains all the possible subsets of the elements. For this case for each element we have 3-1 = 2 options in order to combine, then the number of possible subsets would be equal to the product of possible options and we got:

P({a, b, {a, b}})=

Part b

P({∅, a, {a}, {{a}}}) as we can see this set contains 4 elements ∅,a,{a} and {{a}}

And since the power set contains all the possible subsets of the elements. For this case for each element we have 3-1 = 2 options in order to combine, then the number of possible subsets would be equal to the product of possible options and we got:

P({∅, a, {a}, {{a}}})=

Part c

P(P(∅)) for this case we need to remember that P(∅) have all the possible subsets empty, but the power set for the empty set is also empty.

P(∅)={∅}

And we see that this last one set have just one element.

For this special case we have again 2 options since we can have the element in the subset or the element that is not on the subset, so then we have this:

P(P(∅))=P(P{∅})=2

the one on the bottom left

step-by-step explanation:

all of the other ones have a specific pattern