indeed...a MEMBER of an equivalence class is called a "representative". we don't need to pick these in a uniform way, but sometimes, they help illuminate the differences between the various equivalences classes.

for this example, we could pick $\displaystyle (\sqrt{a},0)$ as the "designated representative", which would tells us it comes from the level set defined by f(x,y) = a.

the other function i gave you:

g(x,y) = xy

works in much the same way, except now the level-sets are hyperbolas instead of circles.

if instead, our function was:

h(x,y) = x + y, the level-sets would be lines, instead of circles or hyperbolas.

we can do the same thing with a variety of different curve-shapes (parabolas, ellipses, squares) although finding a function that produces the level-set shape we want can take a bit of ingenuity.

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an equivalence relation (on a set S) is often defined (this is the definition you are probably using in your class) as a relation (a subset of SxS) that:

1) is reflexive (a~a, or equivalently: includes the diagonal set ΔS = {(a,a) in SxS: a in S})

2) is symmetric (if a~b then b~a, or equivalently: is symmetric about the diagonal)

3) is transitive (if a~b and b~c, then a~c, equivalently: the triangle of points (a,b),(b,c),(a,c) is in ~)

but there is another way of looking at equivalence relations on S: as PARTITIONS of S. so even though a relation may not LOOK like an equivalence relation, if it chops up the set S into disjoint pieces, its an equivalence.

the general idea is "all elements of an equivalent class share some property, so it doesn't matter which one we use". for our good old friend $\displaystyle f(x,y) = x^2 + y^2$ (this is also known as "the square of the distance from the origin function") if two elements are in the same equivalence class, we know they both lie on the same circle (centered at the origin). so if all we want to do is pick "some point" a fixed distance from the origin, any representative from the equivalence class is as good as any other.