Yes. The meaning of a function is a relation which has second property: if both $(x,y)$ and $(x,z)$ room in the relation climate $y=z$.

You are watching: Every relation is a function.

More to the point, though, the notation $fcolon X o Y$ method three things:

$f$ is a function, which is a relation with this one-of-a-kind property.$operatornamedom(f)=X$, therefore**every**suggest in $x$ appears as the left name: coordinates of part ordered pair in $f$.$operatornamerng(f)subseteq Y$, therefore if $(x,y)in f$ then $yin Y$.

There room contexts wherein we eliminate the second requirement. For example in forcing we frequently want to talk around partial features from a huge domain, so us omit the 2nd requirement. Yet this is normally mentioned clearly in the text.

As Halmos says, view a relationship (in specific a function) in terms of something the

*is*, no something that

*does*.

A relationship is a subset of $X imes Y$. That is all. You choose pairs $(x,y)$ with $xin X,yin Y$. If $(x,y)in R$; then $x$ stands in relation with $y$.

A role is a special type of relation: all aspects of $X$ need to be concerned at least one element of $Y$. Moreover, this facet $xin X$ is regarded is unique, that is, $x$ cannot be associated to more than one aspect of $Y$.

So, for example, a function $f:1,3,6,5\toa,b,c$ is $$f=(1,a),(3,b),(6,b),(5,c)$$

while a relation can be $$R=(1,a),(1,b),(3,b),(3,c)$$

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