Discrete Mathematics with Applications 4th Edition Chapter 7 - Functions - Exercise Set 7.1 - Page 396 50

Answer

$a-\,\,\\ X_{A\cap B}(u)=0 \,\,\,or\,\,\,X_{A\cap B}(u)=1\\ First\,\,case\,:\\ X_{A\cap B}(u)=0\Leftrightarrow u\notin A\cap B\Leftrightarrow u\notin A\,\,or\,\,u\notin B \\ \Leftrightarrow X_{A}(u)=0\,or\,X_{B}(u)=0 \Leftrightarrow X_{A}(u).X_{B}(u)=0=X_{A\cap B}(u) \\ Second\,\,case\,:\\ X_{A\cap B}(u)=1\Leftrightarrow u\in A\cap B\Leftrightarrow u\in A\,\,or\,\,u\in B \\ \Leftrightarrow X_{A}(u)=1\,and\,X_{B}(u)=1 \Leftrightarrow X_{A}(u).X_{B}(u)=1=X_{A\cap B}(u) \\ $ $b-\\ u\in A\cup B\,\,or\,\,u\notin A\cup B \\ X_{A\cup B}(u)=1 \,\,\,or\,\,\,X_{A\cup B}(u)=0\\ u\in A\cup B \Leftrightarrow X_{A\cup B}(u)=1\\ if\,\,u\in A \,\,and\,\,u\notin B \\ X_{A\cup B}(u)=X_{A}(u)+X_{B}(u)-X_{A}(u).X_{B}(u)\\=1+0-0=1\\ if\,\,u\in B\,\,and\,\,u\notin A \\ X_{A\cup B}(u)=X_{A}(u)+X_{B}(u)-X_{A}(u).X_{B}(u)\\=0+1-0=1 \\ if\,\,u\in B\,\,and\,\,u\in A \\ X_{A\cup B}(u)=X_{A}(u)+X_{B}(u)-X_{A}(u).X_{B}(u)\\=1+1-1=1 \\ so\,\,in\,all\,cases:\\ X_{A\cup B}(u)=X_{A}(u)+X_{B}(u)-X_{A}(u).X_{B}(u)=1\\$ $ u\notin A\cup B\Leftrightarrow X_{A\cup B}(u)=0 \\ X_{A}(u)+X_{B}(u)-X_{A}(u).X_{B}(u)=0+0-0=0=X_{A\cup B}(u)\\ $

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