Let $A$ and $B$ be sets, and let $S \subseteq A$ be a subset. Let $F: S\to B$ be a function , and let $G:A\to B$ be an extension of $F$. suppose that $F$ is injective . is $G$ necessarily injective? Give a proof or a counterexample.
I am very confused please help , thanks
Let's make this concrete.
Consider $S= \{1, 2, 3\}$, $A = \{1,2, 3,4, 5\}$ and $B = \{11, 12, 13\}$.
Define $F(i) = i+10$ for $i \in S$. We can extend $F$ to some function $G$. Now, $G$ can take any values for $4, 5$.
Play around with this concrete example and see what you think.
In general, if you're struggling with a problem or some abstract idea, try to make it concrete. Make a simple example and see what the properties are.