Venn diagram of relations with prescribed properties

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Draw the arrow diagram of a relation between a set $A$ of the seven days of the week and a set $B$ of the seven capital vices (wrath, pride, lust, gluttony, greed, sloth, envy) so that:

• (a) the relation is not a function;
• (b) the relation is an injective but not surjective function;
• (c) the relation is a surjective but not injective function;
• (d) the relation is a bijective function.

If needed, you may add or remove some vice in $B$. Explain each drawing. (figure in the original test)

Solution

Call the days $g_1,\dots,g_7$ (domain $A$) and the vices the elements of $B$. The key is to control the arrows leaving $A$ (for the definition of function) and those entering $B$ (for injectivity/surjectivity).

• (a) Not a function: just break uniqueness of the image. E.g. send two arrows from $g_1$ (to wrath and gluttony), or leave $g_2$ with no arrow. Either way $A$ is not the domain of a function.
• (b) Injective not surjective: every day one arrow, all to different vices; with $|B|>7$ (add an eighth vice) at least one vice is not reached. Distinct arrows ⇒ injective; uncovered vice ⇒ not surjective.
• (c) Surjective not injective: reduce $B$ to fewer than $7$ vices (e.g. $6$); every day one arrow and every vice reached (surjective), but by the pigeonhole principle at least two days land on the same vice ⇒ not injective.
• (d) Bijective: $|A|=|B|=7$, every day one arrow to a different vice and all vices reached: a one-to-one correspondence. $\boxed{\ \text{(a) double/missing arrow};\ \text{(b) } |B|=8 \text{ all distinct};\ \text{(c) } |B|=6 \text{ with one coincidence};\ \text{(d) } |B|=7 \text{ all distinct}\ }$