I have a question regarding performing set operations on empty sets.
For example let A = ∅, Let B = {A, ∅}, Let C = {A, B}.
Would B = {∅, ∅} and C = {∅, {∅,∅}}? or would B = {∅} and C = {∅, {∅}} since {∅, ∅} reduces to {∅}?
Now if I wanted to do $A \cup C$ would the answer be $∅ \cup C$ -> C = {∅, {∅}}?
Now what if I wanted to do D = C - B, would the result be {{∅}}?
One last question. If P = {∅}, Q = {P}, R = {∅,P}, S = {∅, P, Q, R};
What would X = {$ x : (x ∈ R) \wedge ( x ⊆ S)$}?
My thoughts are that Q = {{∅}}, R = {∅, {{∅}}} and S = { ∅, {∅}, {{∅}}, {∅, {{∅}}} }
- x ∈ R would mean x = ∅ and x = {{∅}}
- x ⊆ S would mean x is a set containing any combination of ∅, {∅}, {{∅}}, {∅, {{∅}}} as members
Therefore X = {{{∅}}} as {{∅}} is the only member that fits both those conditions?
I've tried looking at some lectures online on set theory but I want to make sure my understanding is correct so far.
On a more general note, I am also wondering if lets say D = {1, 2} and E = {D, 5}. Would E = {1,2,5} or E = {{1,2},5}?
Thanks