Answer:
a) True
b) True
c) False
d) True
e) True
f) True
g) True
Step-by-step explanation:
Let A be a set. a belongs (is element of) to A if and only if a∈A.
a) The only element of {∅} is ∅, thus ∅∈{∅}.
b) The elements of {∅, {∅}} are ∅ and {∅}, hence ∅ ∈ {∅, {∅}}.
c) The only element of {∅} is ∅. We have that {∅}≠∅ because {∅} has one element but ∅ has 0 elements. Therefore, {∅} ∉ {∅}, since {∅} is not an element of {∅}.
d) The only element of {{∅}} is {∅}. Hence {∅} ∈ {{∅}}.
e) The elements of {∅, {∅}} are ∅ and {∅}. The only element of {∅} is ∅. Then, every element of {∅} is also an element of {∅, {∅}}. That is, {∅} ⊂ {∅, {∅}}.
f) The elements of {∅, {∅}} are ∅ and {∅}. The only element of {{∅}} is {∅}. Then, every element of {{∅}} is also an element of {∅, {∅}}. That is, {{∅}} ⊂ {∅, {∅}}.
g) The only element of {{∅}, {∅}}={{∅}} is {∅}. The only element of {{∅}} is {∅}. Then, every element of {{∅}} is also an element of {{∅}}. That is, {{∅}} ⊂ {{∅}, {∅}}={{∅}}.
