Determine whether these statements are true or false

$\varnothing$ is an empty set, in turn, an empty set does not contain any elements (or sets). X is a subset of Y, and if every element of X is also an element of Y. ${\displaystyle X\subseteq Y}$, ${\displaystyle X\subset Y}$This means that X is a subset of Y, but X is not the same set of Y. x is an element of Y. ${\displaystyle x\in Y}$
Solution
a) Given: $\varnothing \in \{\varnothing \}$
$\{\varnothing \}$ represent the set containg only the empty set and thus the emply set is an element of $\{\varnothing \}$, which means that the given statement is true.
b) Given: $\varnothing \in \{\varnothing ,\{\varnothing \}\}$
The given statement means that the empty set is an element of the set containing the element $\varnothing$ and the subset $\{\varnothing \}$
We then note that $\varnothing$ is an element of the set $\{\varnothing ,\{\varnothing \}\}$, which means that the given satement is true.
c) Given: $\{\varnothing \}\in \{\varnothing \}$
The given statement means that the set $\{\varnothing \}$ is an element of the set containg the emply set.
The set containing the empty set does not contain any sets which contain elements. The set conteing the empty set is a set which contains elements, trus the given statement is false.
d) Given: $\{\varnothing \}\in \{\{\varnothing \}\}$
The given statement means that the set $\{\varnothing \}$ is an element of the set containing the set $\{\varnothing \}$
Since $\{\varnothing \}$ is the only set contained in the set $\{\varnothing \}$, the given statement is true.
e) Given $\{\varnothing \}\subset \{\varnothing ,\{\varnothing \}\}$
The given statement means that the set $\{\varnothing \}$ is a subset of the set containing element $\varnothing$ and set $\{\varnothing \}$
The set $\{\varnothing \}$ conteins only the element $\{\varnothing \}$ Since $\varnothing$ is also an element in $\{\varnothing ,\{\varnothing \}\}$, the given statement is true
f) Given: $\{\{\varnothing \}\}\subset \{\varnothing ,\{\varnothing \}\}$
The given statement means that the set $\{\{\varnothing \}\}$ is a subset of the set containing elements $\varnothing$ and $\{\varnothing \}$
The set $\{\{\varnothing \}\}$ contains only the element $\{\varnothing \}$. Since $\{\varnothing \}$ is also an element in $\{\varnothing ,\{\varnothing \}\}$, the given statement is true.
g) Given: $\{\{\varnothing \}\}\subset \{\{\varnothing \},\{\varnothing \}\}$
The given statement means that the set $\{\varnothing \}$ is a subset of the set containg the element $\{\varnothing \}$
The set $\{\varnothing \}$ contains only the element $\{\varnothing \}$ and since $\{\{\varnothing \},\{\varnothing \}\}$ contains only the element $\{\varnothing \}$ as well, the two sets are the same set. The given statement is then false as $\{\{\varnothing \}\}=\{\{\varnothing \},\{\varnothing \}\}$
Note: the statement $\{\{\varnothing \}\}\subset q\{\{\varnothing \},\{\varnothing \}\}$ is true (as the sets are equal).