Answer
f is not one-to-one
f is not onto
Work Step by Step
$F:p(\left \{ a,b,c \right \})\rightarrow \mathbb{Z} \\
F(A) =\,the\,number\,of\,elements\,in\,A.(A\subseteq p\left \{ a,b,c \right \} )\\
A\,function\,\,F: p(\left \{ a,b,c \right \}) \rightarrow \mathbb{Z}\,is\,not\,\,\,one-to-one\,\Leftrightarrow \\
\exists \,\,x_{1}\,and\,x_{2}\,\,in\,\,p(\left \{ a,b,c \right \})\,\,such\,that\,\,
F(x_{1}) = F(x_{2})\,\,and x_{1} \neq x_{2}.\\
we\,see\,that\,f(\left \{ a \right \})=f(\left \{ b \right \})=1\\
and \, \left \{ a \right \}\neq \left \{ b \right \} \\
f\,is\,not\,one-to-one \\
F: p(\left \{ a,b,c \right \}) \rightarrow \mathbb{Z}\, is\,\,not\,\,onto\,\Leftrightarrow \,\\
\exists y\,in\,\mathbb{Z} such\,that\,\forall x \in p(\left \{ a,b,c \right \}), F(x) \neq y.\\
for\,example\,5\in\mathbb{Z}\,and\,there\,is\,no\,x\in p(\left \{ a,b,c \right \})such\,that\,\\
f(x)=5 \\(as\,the\,maximum\,number\,of\,elements\,that\,can\,be\,in\,a\,power\,set\,is
\,3)
\\ f\,is\,not\,onto
$
