True or false on membership and inclusion

Statement

Decide whether each statement is true or false.

(i) $3 \in \{3\}$ (ii) $4 \in \varnothing$ (iii) $\varnothing \subseteq \{1,2,3\}$ (iv) $\{a\} \subseteq \{a,b\}$ (v) $\{1,2\} \in \{1,2,3\}$ (vi) $\varnothing \in \{\varnothing\}$ (vii) $\{\varnothing\} \subseteq \{\varnothing\}$ (viii) $a \in \{b,\{a\}\}$

Solution

(i) T: $3$ is the only element of $\{3\}$. (ii) F: the empty set has no elements. (iii) T: the empty set is a subset of every set. (iv) T: the only element $a$ of $\{a\}$ belongs to $\{a,b\}$. (v) F: $\{1,2\}$ is a subset of $\{1,2,3\}$, not an element of it (the elements are $1,2,3$). (vi) T: $\varnothing$ is exactly the only element of $\{\varnothing\}$. (vii) T: every set is a subset of itself. (viii) F: the elements of $\{b,\{a\}\}$ are $b$ and $\{a\}$, not $a$.