Chapter 6 - Set Theory - Exercise Set 6.3 - Page 372: 6

$For\,\,all\,\,sets\,\,A,\,B,\,and\,\,C\,\\ A\cap (A\cup B)=A\\ this\,\,is\,\,\,\,true:\\ to\,prove\,this\,we\,must\,prove\,:\\ 1-A\cap (A\cup B)\subseteq A\\ 2-A\subseteq A\cap (A\cup B)\\\ Proof\,\,of\,\,1:\\ x\in A\cap (A\cup B)\Rightarrow x\in A\,and\,x\in (A\cup B)\\ \Rightarrow so\,\,x\in A\\ \because x\in A\cap (A\cup B)\Rightarrow x\in A \\ \therefore A\cap (A\cup B)\subseteq A\\ $ $proof\,\,of\,\,2:\\ x\in A \Rightarrow x\in A\cup B \\ (by\,\,def.\,\,of\,union)\\ \Rightarrow x\in A\,\,and\,\,x\in A\cup B \\ \Rightarrow x\in A\cap (A\cup B) \\ (by\,\,def.\,\,of\,intersection)\\ \because x\in A \Rightarrow x\in A\cap (A\cup B) \\ \therefore so\,\,A\subseteq A\cap (A\cup B)\\ from\,\,1,2\,\,\\ A\cap (A\cup B)=A\\$

$For\,\,all\,\,sets\,\,A,\,B,\,and\,\,C\,\\ A\cap (A\cup B)=A\\ this\,\,is\,\,\,\,true:\\ to\,prove\,this\,we\,must\,prove\,:\\ 1-A\cap (A\cup B)\subseteq A\\ 2-A\subseteq A\cap (A\cup B)\\\ Proof\,\,of\,\,1:\\ x\in A\cap (A\cup B)\Rightarrow x\in A\,and\,x\in (A\cup B)\\ \Rightarrow so\,\,x\in A\\ \because x\in A\cap (A\cup B)\Rightarrow x\in A \\ \therefore A\cap (A\cup B)\subseteq A\\ $ $proof\,\,of\,\,2:\\ x\in A \Rightarrow x\in A\cup B \\ (by\,\,def.\,\,of\,union)\\ \Rightarrow x\in A\,\,and\,\,x\in A\cup B \\ \Rightarrow x\in A\cap (A\cup B) \\ (by\,\,def.\,\,of\,intersection)\\ \because x\in A \Rightarrow x\in A\cap (A\cup B) \\ \therefore so\,\,A\subseteq A\cap (A\cup B)\\ from\,\,1,2\,\,\\ A\cap (A\cup B)=A\\ $