how would prove this using for set operations is..... idempotent laws associa........

Hello, npm1!

Quote:

Prove using Set operations:

. . $\bigg[(X \cup Y) \cap (X \cup Y')\bigg] \cap \bigg[(X' \cup Y) \cap (X' \cup Y')\bigg] \;=\;\emptyset$

. . $\begin{array}{ccc}\bigg[(X \cup Y) \cap (X \cup Y')\bigg] \cap \bigg[(X' \cup Y) \cap (X' \cup Y')\bigg] & \text{Given} \\ \\ \bigg[X \cup (Y \cap Y')\bigg] \cap \bigg[X' \cup (Y \cap Y')\bigg] & \text{Distr.} \\ \\ \bigg[X \cup\: \emptyset\bigg] \cap \bigg[X' \cup \:\emptyset\bigg] & A \cap A' \:=\:\emptyset \\ \\ X \cap X' & A \cup \emptyset \:=\:A \end{array}$

. . . . . . . . . . . . . . . . $\begin{array}{ccccccc}\emptyset & \qquad\qquad & \qquad\qquad\;\; & A \cap A' \:=\:\emptyset\end{array}$