[Math] Find four binary relations from $\mathbf{\{a,b\}}$ to $\mathbf{\{x,y\}}$ that are not functions from $\mathbf{\{a,b\}}$ to $\mathbf{\{x,y\}}$

Question

Question: Find four binary relations from $\mathbf{\{a,b\}}$ to $\mathbf{\{x,y\}}$ that are not functions from $\mathbf{\{a,b\}}$ to $\mathbf{\{x,y\}}$.

Thoughts: I know that a relation $\mathbf R$ from $\mathbf A$ to $\mathbf B$ is the subset of $\mathbf{A \ast B}$, so the possible relations are $\mathbf{2^4 \to 16}$. However, I am stumped when trying to apply the rules for functions to this question. Care to explain?

To further elaborate, is there a general – intuitive – way of tackling the problem than just writing them all down?

Answer 1

(Converting Paul Sinclair's comment to an answer.)

You appear to overlooking one of the two requirements of a function. Recall that a function can only assign each element of its domain to a single element of the codomain. Note that in the examples presented, they are mapping $a$ to two elements of $\{x,y\}$. Breaking either the "must map all elements of the domain" or the "each element only maps to one image" rule will prevent it from being a function.