Answer
$for\,\,all\,sets\,\,A\,\,, A\cup \varnothing =A \\
to\,\,prove\,\,this\,\,we\,must\,prove\,\,that\,\,\\
1-A\cup \varnothing \subseteq A \\
2-A \subseteq A\cup \varnothing \\
proof\,\,of\,\,1:\\
x\in A\cup \varnothing \overset{def.of union}{\rightarrow} x\in A\,\,or\,\,x\in\varnothing \\
since\,\,\varnothing\,\,is\,the\,empty\,set\rightarrow x\,must\,belong\,to\,A\rightarrow x\in A\\
so\,\,A\cup \varnothing \subseteq A \\
proof\,\,of\,\,2:\\
x\in A \overset{def.of\,union}{\rightarrow}x\in A\cup \varnothing\\
so\,\,A \subseteq A\cup \varnothing \\
from\,1\,,2\, \,\,, A\cup \varnothing =A$
Work Step by Step
$for\,\,all\,sets\,\,A\,\,, A\cup \varnothing =A \\
to\,\,prove\,\,this\,\,we\,must\,prove\,\,that\,\,\\
1-A\cup \varnothing \subseteq A \\
2-A \subseteq A\cup \varnothing \\
proof\,\,of\,\,1:\\
x\in A\cup \varnothing \overset{def.of union}{\rightarrow} x\in A\,\,or\,\,x\in\varnothing \\
since\,\,\varnothing\,\,is\,the\,empty\,set\rightarrow x\,must\,belong\,to\,A\rightarrow x\in A\\
so\,\,A\cup \varnothing \subseteq A \\
proof\,\,of\,\,2:\\
x\in A \overset{def.of\,union}{\rightarrow}x\in A\cup \varnothing\\
so\,\,A \subseteq A\cup \varnothing \\
from\,1\,,2\, \,\,, A\cup \varnothing =A$