$\displaystyle \displaystyle \text{let} \ x \in (A \cap B^{c}) \cup (B \cap A^{c}) $

$\displaystyle \displaystyle \text{then} \ x \in (A \cap B^{c}) \ \text{or} \ x \in (B \cap A^{c}) $

$\displaystyle \text{so} \ (x \in A \ \text{and} \ x \in B^{c}) \ \text{or} \ (x \in B \ \text{and} \ x \in A^{c}) $

$\displaystyle \Big((x \in A \ \text{and} \ x \in B^{c}) \ \text{or} \ x \in B \Big) \ \text{and} \ \Big((x \in A \ \text{and} \ x \in B^{c}) \ \text{or} \ x \in A^{c} \Big)$

$\displaystyle \Big((x \in A \ \text{or} \ x\in B) \ \text{and} \ (x \in B^{c} \ \text{or} \ x \in B) \Big) \ \text{and} \ \Big((x \in A \ \text{or} \ x \in A^{c}) \ \text{and} \ (x \in B^{c} \ \text{or} \ x \in A^{c}) \Big) $

$\displaystyle \Big((x \in (A \cup B) \cap U \Big) \ \text{and} \ \Big(x \in U \cap (B^{c} \cup A^{c}) \Big) $

$\displaystyle x \in (A \cup B) \ \text{and} \ x \in (A \cap B)^{c} $

$\displaystyle x \in (A \cup B) \cap (A \cap B)^{c} $