Check if a relation on a set is a function

Question

What do I need to look for in order to tell if a relation on a set is a function?

Can somebody provide some advice?

For example, the relation is defined by $H$ on $A \times \mathcal P(A)$ for $a ∈ A$ and $B ⊆ A$, $a\, R\, B$ iff $a ∈ B$.

Where $A =\{0,1\}$ does this mean H is a function?

Answer 1

A binary relation on a set $A$ is a collection of ordered pairs of elements of $A$. An element of $A$ can be related to many elements of $A$.

A function is a relation between a set $A$ and a set $B$ with the property that each element of $A$ is related to exactly one element of $B$.

Having said that, all functions are relations but not all relations are functions.

A relation on a set $A$ is a function iff every element of $A$ is related or mapped to only one element. If you find that an element of $A$ is related to more than one element then the given relation cannot be a function.

Answer 2

• For every $a\in A$ can you find a $B\in\wp A$ satisfying $aHB$?
• Is this $B$ unique in satisfying this condition?

If (and only if) the answer on both questions is 'yes' then $H$ is a function.