Building a partial injective relation Question : A Partial Injective Relation from $A \rightarrow B$ is maximal if its graph of an injection function from $A$ to $B$ or the graph of an injection function from $B$ to $A$. 
Example: $A =[a,b,c]$ and $B=[1,2,3,4]$. 
Build a partial injection relation. Are there 3-4 elements or a lot of elements?
Attempt: 
Definition 5.4.5 states that a function $f: X \rightarrow Y$ with the property $ \forall x_1, x_2 \in X)[ x_1 \neq x_2] \rightarrow f(x_1) \neq f(x_2)$ is an injection of $X$ into $Y$
We're using the contrapositive of the injection definition which is  $ \forall x_1, x_2 \in X)[ x_1 = x_2] \rightarrow f(x_1) = f(x_2)$
Suppose $ f: b \rightarrow a$ is a partial injection. 
From the example, we have $A =[a,b,c]$ and $B = [1,2,3,4]$. This is a partial injective relation if we have $S=[(a,1),(b,2),(c,3)]$. There are only three elements because from $B$, the number $4$ isn't matching anything on $A$. 
So if I construct a bijection map $f: B \rightarrow A$, I would have $f(1) = a$, $f(2) = b$, and $f(3) =c$. I won't have anything equal to 4 because set $A$ only has three elements and set $B$ has four elements. Therefore, if I take the bijection, I would have three elements all together. 

There are only injections from the smaller set to the larger, namely
$$
f: A \to B
$$
There are $4$ choices for the value $f(a)$, $3$ choices for $f(b)$, and $2$ choices for $f(c)$ for a total of $4 \cdot 3 \cdot 2 = 24$ possible injective functions.  Each of these gives a distinct maximal injective relation, consisting of the pairs
$$
\big\{ (a, f(a)), (b, f(b)), (c, f(c)) \big\}.
$$
