Prove properties of set equality

Prove the following.

Proof. To show two sets are equal, we want to show that each is a subset of the other (i.e., we want to show that $\{ a,a \} \subseteq \{ a \}$ and $\{ a \} \subseteq \{ a, a \}$ ).
First, $\{ a \} \subseteq \{ a,a \}$ since the only element of $\{ a \}$ is $a$ , and we have $a \in \{ a,a \}$ .
Second, the only element of $\{ a, a \}$ is $a$ , and we have $a \in \{ a \}$ . Hence, $\{ a,a \} \subseteq \{ a \}$ . Thus, $\{ a,a \} = \{ a \}$ . ∎
Proof. Again, we want to show $\{ a,b \} \subseteq \{b,a \}$ and $\{ b,a \} \subseteq \{ a,b \}$ .
First, $\{ a,b \} \subseteq \{ b,a \}$ since the elements of $\{ a,b \}$ are $a$ and $b$ , and $a, b \in \{ b, a \}$ .
Second, the elements of $\{ b,a \}$ are $a$ and $b$ . Since $a,b \in \{a, b\}$ we have every element of $\{ b, a \}$ is in $\{ a,b \}$ ; thus, $\{ b,a \} \subseteq \{a,b \}$ .
Therefore, $\{ a,b \} = \{ b,a \}$ . ∎
Proof. $(\Rightarrow)$ Assume $\{ a \} = \{ b,c \}$ . Since $\{ b,c \} = \{ a \}$ , we must have $\{ b,c \} \subseteq \{ a \}$ ; hence, every element of $\{ b,c \}$ must be contained in $\{ a \}$ . This means that both $b$ and $c$ are in $\{ a \}$ . Since $a$ is the only element of $\{ a \}$ , we must have $a = b = c$ .
$(\Leftarrow)$ Conversely, assume $a = b =c$ . Then $\{ b,c \} = \{ a,a \}$ and from part (a) we know $\{ a \} = \{ a,a \}$ ; hence $\{a \} = \{ b,c \}$ .∎

Stumbling Robot