(1)

\begin{align*} \emptyset^{c} & =\left\{ x;x\in X\nrightarrow x\in\emptyset\right\} \\ & =\left\{ x;x\in X\land x\notin\emptyset\right\} \\ & =\left\{ x;x\in X\right\} \\ & =X \end{align*}

(2)

\begin{align*} X^{c} & =\left\{ x;x\in X\nrightarrow x\in X\right\} \\ & =\left\{ x;x\in X\land x\notin X\right\} \\ & =\emptyset \end{align*}

(3)

\begin{align*} A\cup\emptyset & =\left\{ x;x\in A\lor x\in\emptyset\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(4)

\begin{align*} A\cap\emptyset & =\left\{ x;x\in A\land x\in\emptyset\right\} \\ & =\left\{ x;x\in\emptyset\right\} \\ & =\emptyset \end{align*}

(5)

\begin{align*} A\cup X & =\left\{ x;x\in A\lor x\in X\right\} \\ & =\left\{ x;x\in X\right\} \\ & =X \end{align*}

(6)

\begin{align*} A\cap X & =\left\{ x;x\in A\land x\in X\right\} \\ & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(7)

\begin{align*} A & =\left\{ x;x\in A\right\} \\ & =A \end{align*}

(8)

\begin{align*} A\cup A^{c} & =A\cup\left(X\setminus A\right)\\ & =\left\{ x;x\in A\lor\left(x\in X\nrightarrow x\in A\right)\right\} \\ & =\left\{ x;x\in A\lor\left(x\in X\land x\notin A\right)\right\} \\ & =\left\{ x;x\in A\lor x\in X\right\} \\ & =X \end{align*}

(9)

\begin{align*} A\cap A^{c} & =A\cap\left(X\setminus A\right)\\ & =\left\{ x;x\in A\land\left(x\in X\nrightarrow x\in A\right)\right\} \\ & =\left\{ x;x\in A\land\left(x\in X\land x\notin A\right)\right\} \\ & =\left\{ x;x\in\emptyset\land x\in X\right\} \\ & =\left\{ x;x\in\emptyset\right\} \\ & =\emptyset \end{align*}

(10)

\begin{align*} A=B & \Leftrightarrow A\subseteq B\land A\supseteq B\\ & \Leftrightarrow\left(x\in A\rightarrow x\in B\right)\land\left(x\in A\leftarrow x\in B\right)\\ & \Leftrightarrow\left(x\in A^{c}\leftarrow x\in B^{c}\right)\land\left(x\in A^{c}\rightarrow x\in B^{c}\right)\\ & \Leftrightarrow A^{c}\supseteq B^{c}\land A^{c}\subseteq B^{c}\\ & \Leftrightarrow A^{c}=B^{c} \end{align*}