Question

Determine whether these statements are true or false. a) ∅ ∈ {∅} b) ∅ ∈ {∅, {∅}} c) {∅} ∈ {∅} d) {∅} ∈ {{∅}} e) {∅} ⊂ {∅, {∅}} f ) {{∅}} ⊂ {∅, {∅}} g) {{∅}} ⊂ {{∅}, {∅}}

Answer 1

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, {{∅}} ⊂ {{∅}, {∅}}={{∅}}.