Question

Determine whether the statements below are true or false.
(a)Ø ϵ {Ø}
(b) Ø ϵ {Ø,{Ø}}
(c){Ø} ϵ {Ø}
(d) {Ø} ϵ {{Ø}}
(e) {Ø} ⊆ {Ø,,{Ø}}
(f){{Ø}} ⊆ {Ø,{Ø}}
(g) {{Ø}} ⊆ {{Ø},{Ø}}

Answer 1

a)True

Only one element of $\{\varnothing\}$ is $\varnothing$, so $\varnothing\in\{\varnothing\}$

b)True

Elements of $\{\varnothing,\{\varnothing\}\}$ are $\varnothing$ and $\{\varnothing\}$, so $\varnothing\in\{\varnothing,\{\varnothing\}\}$

c)False

Only one element of $\{\varnothing\}$ is $\varnothing$, so $\{\varnothing\}$ is not an element of $\{\varnothing\}$, $\{\varnothing\}\not\in\{\varnothing\}$

d)True

Only one element of $\{\{\varnothing\}\}$ is $\{\varnothing\}$, so $\{\varnothing\}\in\{\{\varnothing\}\}$

e)True

Only one element of $\{\varnothing\}$ is $\varnothing$, and elements of $\{\varnothing,\{\varnothing\}\}$ are $\varnothing$ and $\{\varnothing\}$. So every element of $\{\varnothing\}$ is an element of $\{\varnothing,\{\varnothing\}\}$, that is $\{\varnothing\}\subseteq\{\{\varnothing,\{\varnothing\}\}$

f)True

Only one element of $\{\{\varnothing\}\}$ is $\{\varnothing\}$ , and elements of $\{\varnothing,\{\varnothing\}\}$ are $\varnothing$ and $\{\varnothing\}$ . So every element of $\{\{\varnothing\}\}$ is an element of $\{\varnothing,\{\varnothing\}\}$ , that is $\{\{\varnothing\}\}\subseteq\{\varnothing,\{\varnothing\}\}$

g)True

Only one element of $\{\{\varnothing\}\}$ is $\{\varnothing\}$, and elements of $\{\{\varnothing\},\{\varnothing\}\}$ are $\{\varnothing\}$ and $\{\varnothing\}$. So every elemen of $\{\{\varnothing\}\}$ is an element of $\{\{\varnothing\},\{\varnothing\}\}$, that is $\{\{\varnothing\}\}\subseteq\{\{\varnothing\},\{\varnothing\}\}$