Answer
$U^{c}=\varnothing \\
To\,\,prove\,\,that\,\,a\,\,set\,\,U^{c}\,\,is\,\,equal\,\,to\,\,the\,\,empty\,\,set\,\, \varnothing ,\\ prove\,\,that\,\,U^{c}\,\,has\,\,no\,\,elements. \\ To\,\,
do\,\,this, suppose\,\,U^{c} \,\,has\,\,an\,element\,\,and\,\,derive\,\,a\,\,contradiction \\
suppose\,\,x\in U^{c}\,\,(by\,\,def.\,\,of\,\,complement )\Rightarrow x\notin U \\
(this\,\,is\,\,a\,\,contradiction)\\
because\,\,by\,\,def.\,\,\,\,a\,\,universal\,set\,\,contain\,all\,elements\,\\
so\,\,U^{c}=\varnothing
$
Work Step by Step
$U^{c}=\varnothing \\
To\,\,prove\,\,that\,\,a\,\,set\,\,U^{c}\,\,is\,\,equal\,\,to\,\,the\,\,empty\,\,set\,\, \varnothing ,\\ prove\,\,that\,\,U^{c}\,\,has\,\,no\,\,elements. \\ To\,\,
do\,\,this, suppose\,\,U^{c} \,\,has\,\,an\,element\,\,and\,\,derive\,\,a\,\,contradiction \\
suppose\,\,x\in U^{c}\,\,(by\,\,def.\,\,\,complement )\Rightarrow x\notin U \\
(this\,\,is\,\,a\,\,contradiction)\\
because\,\,by\,\,def.\,\,of\,\,a\,\,universal\,set\,\,contain\,all\,elements\,\\
so\,\,U^{c}=\varnothing $
