# Difference of mapsto and right arrow

Could someone please explain to me what is the difference in the two arrows$$\rightarrow$$ and $$\mapsto$$ For example in Probability wih Martingales (Willams)

Thank you.

The arrow $\rightarrow$ denotes a mapping between two sets. The $\mapsto$ is telling you what it does to each element of the set.

We can say:$$f:\{1,2,3\}\to\{4, 5, 6\}$$ This is a function from the set $$\{1,2,3\}$$ to the set $$\{4, 5, 6\}$$. Both things on the right and left of the arrow are sets.

But, if we say: $$1\mapsto5$$ This says that when we put $$1$$ into the function/relation/whatnot, we get out a $$5$$. Both things on the left and right of the arrow are elements of the the domain/codomain.

This is not to say that we couldn't have a function like:* $$f:\{\{1, 2\},\{3, 4\},\{5, 6\}\}\to\{4, 5, 6\}\text{ such that:}\\\{1, 2\}\mapsto 4\\\vdots\\\text{etc.}$$ In this case, one of the things on the right/left of the $$\mapsto$$ is a set. We could even make both of the things a set; it doesn't matter for $$\mapsto$$, but $$\to$$ must have sets on both sides. The $$\mapsto$$ is really just specifying what we get out when we put in something.

• Thank you, very nice explanation. – triomphe Aug 22 '13 at 2:07

Typically $\to$ is used when specifying the domain and codomain of an arbitrary function while $\mapsto$ is used in explicit definitions. So $f: A \to B$ means that $f$ is a function with domain $A$ and codomain $B$ (note $B$ might not be the range of $f$). Notice this doesn't define a specific $f$ from $A$ to $B$, but instead could represent any such function.

On the other hand, when I say $f: \mathbb{R} \to \mathbb{R}$ is defined by $f: x \mapsto x^2$, the second arrow simply says that this is the function $f(x) = x^2$.