Some math help please.

[VTBoy]

For each element a in R (Set of Reals), let Aa = { (x,y) in R X R: y = a-x^2}

Prove that {Aa:a in R} is a partition of RXR
And Describe the equivalent relation associated with this partition.

[Severian]

To show it is a partition, we must first show that each (x,y) in R x R is in one of the Aa's. This isn't so bad; let (x,y) in R x R and let a=y+x^2.
Then we must show that the Aa's are disjoint. This also isn't so bad; if (x,y) in Aa1 and Aa2, then y+x^2 = a1 and y+x^2 = a2, so a1=a2, so each point can' t be in two distinct Aa's.

Finally, to describe the equivalence relation, you need to ask yourself what kind of equivalence relation will allow you to say (x1,y1) ~ (x2,y2) iff (x1,y1) and (x2,y2) are in the same Aa (the answer is pretty much given above).

[VTBoy]

To show it is a partition, we must first show that each (x,y) in R x R is in one of the Aa's. This isn't so bad; let (x,y) in R x R and let a=y+x^2.
Then we must show that the Aa's are disjoint. This also isn't so bad; if (x,y) in Aa1 and Aa2, then y+x^2 = a1 and y+x^2 = a2, so a1=a2, so each point can' t be in two distinct Aa's.

Finally, to describe the equivalence relation, you need to ask yourself what kind of equivalence relation will allow you to say (x1,y1) ~ (x2,y2) iff (x1,y1) and (x2,y2) are in the same Aa (the answer is pretty much given above).

thanks for the help