Answer
$A\cup (A\cap B)=A \\
to\,\,prove\,\,this\,\,we\,must\,prove\,\,that\,\,\\
1-A\cup (A\cap B)\subseteq A \\2-A\subseteq A\cup (A\cap B)\\
proof\,of\,\,1:\\suppose\,x\in A\cup (A\cap B)\overset{def.of union}{\rightarrow}x\in A\,\,or\,x\in\left ( A\cap B \right )\overset{def\,of\,inter\! section}{\rightarrow}x\in A\,or\,x\in A\,and\,x\in B(so\,\,in\,both\,cases\,\,x\in\,A)\\so\,\, A\cup (A\cap B)\subseteq A \\$
$proof\,\,of\,\,2:\\
suppose\,x\in\,A\overset{def.of union}{\rightarrow}x\in A\cup (A\cap B)\\
so\,\,A\subseteq A\cup (A\cap B)$
$so\,\,A\cup (A\cap B)=A \\$
Work Step by Step
$A\cup (A\cap B)=A \\
to\,\,prove\,\,this\,\,we\,must\,prove\,\,that\,\,\\
1-A\cup (A\cap B)\subseteq A \\2-A\subseteq A\cup (A\cap B)\\
proof\,of\,\,1:\\suppose\,x\in A\cup (A\cap B)\overset{def.of union}{\rightarrow}x\in A\,\,or\,x\in\left ( A\cap B \right )\overset{def\,of\,inter\! section}{\rightarrow}x\in A\,or\,x\in A\,and\,x\in B(so\,\,in\,both\,cases\,\,x\in\,A)\\so\,\, A\cup (A\cap B)\subseteq A \\$
$proof\,\,of\,\,2:\\
suppose\,x\in\,A\overset{def.of union}{\rightarrow}x\in A\cup (A\cap B)\\
so\,\,A\subseteq A\cup (A\cap B)$
$so\,\,A\cup (A\cap B)=A \\$
