Answer
a) True
b) True
c) False
d) True
e) True
f) True
g) False
Work Step by Step
a) $\phi$ $\in$ {$\phi$}
{$\phi$} represent the set containing only one element i.e. the empty set.
Thus empty set is an element of {$\phi$}.
Answer is true.
b)$\phi$ $\in$ { $\phi $, { $\phi$ }}
Statement means that $\phi$ is an element of { $\phi $, { $\phi$ }}.
Clearly $\phi$ is an element of { $\phi $, { $\phi$ }}.
Answer is true.
c){ $\phi$ } $\in$ { $\phi$ }
This statement means that{ $\phi$ } is an element of { $\phi$ }. SInce { $\phi$ } contains on $\phi$.
Hence it cannot contain { $\phi$ }.
Answer is false.
d) { $\phi$ } $\in$ {{ $\phi$ }}
Since {{ $\phi$ }} contains the element { $\phi$ }.
Answer is true.
e){$\phi$} $\subset$ { $\phi $, { $\phi$ }}
Since {$\phi$} contains only the element $\phi$ and $\phi$ is an element in every set. Hence , $\phi$ is an element of { $\phi $, { $\phi$ }}.
Thus, any element of { $\phi$ } is also an element in { $\phi $, { $\phi$ }}.
Answer is true.
f) { {$\phi$} } $\subset$ { $\phi $, { $\phi$ }}
Since { {$\phi$} } contains only the element {$\phi$} and { $\phi$ } is also an element in { $\phi $, { $\phi$ }}.
Answer is true.
g) {{ $\phi$ }} $\subset$ {{ $\phi$ } , { $\phi$ }}
The set {{ $\phi$ }} contains only the element { $\phi$ } as well as the set {{ $\phi$ } , { $\phi$ }} contains only the element { $\phi$ }.
Thus, the two sets are equal.
therefore, the statement will be {{ $\phi$ }} $\subseteq$ {{ $\phi$ } , { $\phi$ }}.
Answer is false.
