Chapter 7 - Functions - Exercise Set 7.1 - Page 396: 41

$If\,A \subseteq B\,\,then\,F(A) \subseteq F(B).\\ assume\,y\in F(A)\,by\,def.\,of\,F(A)\,\\ there\,exists\,\,x\in A\,such\,that\,\\ F(x)=y \\ \because A\subseteq B\,,x\in A\therefore x\in B\\ so\,there\,exist\,some\,element\,x\in B\,such\,that\\ F(x)=y\,\,\,\Rightarrow\,\,y\in F(B) \\ so\, \because y\in F(A)\Rightarrow y\in F(B) \\ \therefore F(A)\subseteq F(B) \\$

$If\,A \subseteq B\,\,then\,F(A) \subseteq F(B).\\ assume\,y\in F(A)\,by\,def.\,of\,F(A)\,\\ there\,exists\,\,x\in A\,such\,that\,\\ F(x)=y \\ \because A\subseteq B\,,x\in A\therefore x\in B\\ so\,there\,exist\,some\,element\,x\in B\,such\,that\\ F(x)=y\,\,\,\Rightarrow\,\,y\in F(B) \\ so\, \because y\in F(A)\Rightarrow y\in F(B) \\ \therefore F(A)\subseteq F(B) \\$