Question

{∅,{∅},{b},{{b}},{{{b}}},{∅,b},{∅,{b}},{∅,{{b}}},{b,{b}},{b.{{b}}},{{b},{{b}}}, {∅,b,{b}},{∅,b,{{b}}},{∅,{b},{{b}}},{b,{b},{{b}}},{∅,b{b},{{b}}}}

is this a power set of a set?

Answer 1

It is known that a set S can be reffered as the power set of the other set s1 if and only if the given set s includes in it , all the subsets of the other set s1(original set).

So, According to this rule we can say that , Yes, the set S1 (let)= {∅,{∅},{b},{{b}},{{{b}}},{∅,b},{∅,{b}},{∅,{{b}}},{b,{b}},{b,{{b}}},{{b},{{b}}}, {∅,b,{b}},{∅,b,{{b}}},{∅,{b},{{b}}},{b,{b},{{b}}},{∅,b{b},{{b}}}} is a power set of the original set S = {∅, b , {b} , {{b}} }

Explanation :

Let S = {∅, b , {b} , {{b}} }.

and S1 = {∅,{∅},{b},{{b}},{{{b}}},{∅,b},{∅,{b}},{∅,{{b}}},{b,{b}},{b,{{b}}},{{b},{{b}}}, {∅,b,{b}},{∅,b,{{b}}},{∅,{b},{{b}}},{b,{b},{{b}}},{∅,b{b},{{b}}}}

so for s1 to be power set of s , the term of the set s1 with largest numbers of elements must be equal to s, Which holds true here. i.e

S = {∅, b , {b} , {{b}} } = largest subset of the set s1.

And s1 must contain all the subsets of the S.

now, let us find the subsets of the set = S = {∅, b , {b} , {{b}} } as follows :

• if we take no elements from s then the subset formed is : (i) ∅

• if we take one elements one by one from set S, then the subset formed are as follows :

(ii) {∅}

(iii) {b}

(iv) {{b}}

(v) {{{b}}}

• Now, If we take two elements together from the set S to form subsets , then we get the following subsets:

(vi) {∅,b}

(vii) {∅,{b}}

(viii) {∅,{{b}}}

(ix) {b,{b}}

(x) {b,{{b}}}

(xi) {{b},{{b}}}

• Now, If we take three elements together from the set S to form subsets , then we get the following subsets:

(xii) {∅,b,{b}}

(xiii) {∅,b,{{b}}}

(xiv) {∅,{b},{{b}}}

(xv) {b,{b},{{b}}}

• Now, If we take all elements together from the set S to form subset , then we get the following subset:

(xvi) {∅,b,{b},{{b}}}

• now combining them all from (1) to (xvi) we get the power set of S ={∅,{∅},{b},{{b}},{{{b}}},{∅,b},{∅,{b}},{∅,{{b}}},{b,{b}},{b,{{b}}},{{b},{{b}}}, {∅,b,{b}},{∅,b,{{b}}},{∅,{b},{{b}}},{b,{b},{{b}}},{∅,b{b},{{b}}}}, Which is equal to the set S1.

Hence we can conclude that the given set {∅,{∅},{b},{{b}},{{{b}}},{∅,b},{∅,{b}},{∅,{{b}}},{b,{b}},{b,{{b}}},{{b},{{b}}}, {∅,b,{b}},{∅,b,{{b}}},{∅,{b},{{b}}},{b,{b},{{b}}},{∅,b{b},{{b}}}} is the power set of the original set S = {∅, b , {b} , {{b}} }