Injective, surjective, bijective functions from arrow diagrams

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For each of the following arrow diagrams between the set $A=\{\text{Alice, Beth, Cara, Dana, Eve}\}$ and a set of boys, decide whether the relation is a function and, if so, whether it is injective, surjective or bijective. (figures in the original test)

• (a) $B=\{$Alex, Bob, Carl, Dave, Eli, Fred$\}$; arrows: Alice→Alex, Beth→Bob, Cara→Carl, Dana→Dave, Eve→Fred.
• (b) same $B$; arrows: Alice→Alex, Alice→Carl, Beth→Alex, Dana→Dave, Eve→Fred (Cara with no arrow).
• (c) $B=\{$Alex, Bob, Carl, Dave, Eli$\}$; arrows: Alice→Alex, Beth→Bob, Cara→Carl, Dana→Dave, Eve→Eli.
• (d) $B=\{$Alex, Bob, Carl, Dave$\}$; arrows: Alice→Alex, Beth→Bob, Cara→Bob, Dana→Dave, Eve→Carl.

Solution

A function assigns to every element of $A$ exactly one element of $B$. It is injective if different elements have different images; surjective if every element of $B$ is the image of at least one element; bijective if both injective and surjective.

• (a) Every girl has exactly one arrow, all to distinct boys: an injective function; but Eli is not reached, so not surjective.
• (b) Alice has two arrows and Cara none: not a function.
• (c) Every girl one arrow to distinct boys, and all five boys are reached: a bijective function.
• (d) Every girl one arrow (function); Beth and Cara both map to Bob → not injective; all four boys are reached → surjective. $\boxed{\ (a)\ \text{injective not surjective};\ \ (b)\ \text{not a function};\ \ (c)\ \text{bijective};\ \ (d)\ \text{surjective not injective}\ }$