Injective non-surjective function with Venn diagram and graph

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Represent, with a Venn diagram (sets and arrows) and also with a graph in the Cartesian plane, a function that is injective but not surjective linking a set of $4$ elements to a set of $5$ elements.

Solution

Conditions. A function from $A$ (domain) to $B$ (codomain) is:

• injective if distinct elements have distinct images (no “double” incoming arrow);
• surjective if every element of $B$ is the image of at least one element of $A$.

With $|A|=4$ and $|B|=5$ the function cannot be surjective: the $4$ arrows reach at most $4$ of the $5$ elements of $B$, so at least one stays uncovered. It is enough to pick $4$ all-distinct images to get an injective, non-surjective function. Example: $A=\{a,b,c,d\}$, $B=\{1,2,3,4,5\}$, with $a\mapsto 1,\ b\mapsto 2,\ c\mapsto 3,\ d\mapsto 4$; the element $5$ is nobody’s image.

Cartesian graph. Placing $A=\{1,2,3,4\}$ on the $x$-axis and reading the images on the $y$-axis, the same function is the point cloud $(1,1),(2,2),(3,3),(4,4)$: no $y$-value repeats (injective), but the codomain value $5$ never appears (not surjective). The image set is $\{1,2,3,4\}\subsetneq B$.