Equivalence Relations for Partition on R^3?

[chocolatelover]

Homework Statement

Suppose that we partition R^3 into horizontal planes. What equivalence relation is associated with this partition? Suppose that we partition R^3 into concentric spheres, centered at (0,0,0). What equivalence relation is associated with this partition?

Homework Equations

The Attempt at a Solution

Since it is on R^3, I know that I need to come up with a partition that has an x, y and z coordinate, right?

Could the equivalence relation be (x,y,z)~(a,b,c) if and only if x^2=a^2?

For the second one, could it be something like (x,y,z)~(0,0,0) if and only if x^2=a^2

Thank you very much

 

[Mathdope]

In both cases equivalent points lie on the same plane (sphere). What are the equations of these planes (spheres)?

 

[HallsofIvy]

There are two questions here. Which you talking about?

Suppose that we partition R^3 into horizontal planes. What equivalence relation is associated with this partition?

The equation of any horizontal plane is z= z₀. Okay, what must be true of (x₁, y₁, z₁) and (x₂, y₂, z₂) in order that they be on the same plane?

Suppose that we partition R^3 into concentric spheres, centered at (0,0,0). What equivalence relation is associated with this partition?

A sphere centered at (0,0,0) has equation x²+ y²+ z²= R². What must be true of (x₁, y₁, z₁) and (x₂, y₂, z₂) if they lie on the same sphere?

 

[chocolatelover]

Thank you very much

Suppose that we partition R^3 into horizontal planes. What equivalence relation is associated with this partition?

The equation of any horizontal plane is z= z0. Okay, what must be true of (x1, y1, z1) and (x2, y2, z2) in order that they be on the same plane?

Z1 and Z2 have to be the same, right? If this is the case, would the partition be something like {{x,y1, n}, {w,y,n}}


Suppose that we partition R^3 into concentric spheres, centered at (0,0,0). What equivalence relation is associated with this partition?

A sphere centered at (0,0,0) has equation x2+ y2+ z2= R2. What must be true of (x1, y1, z1) and (x2, y2, z2) if they lie on the same sphere? Doesn't the z coordinate have to be the same?

Thank you

 

[HallsofIvy]

A sphere centered at (0,0,0) has equation x²+ y²+ z²= R^{2[\sup]. What must be true of (x₁, y₁, z₁) and (x₂, y₂, z₂) if they lie on the same sphere?
Doesn't the z coordinate have to be the same?}

^{No, that was the problem before with planes. If (x₁, y₁, z₁) and (x₂, y₂, z₂) lie on the same sphere then they must both satisfy the equation of that sphere: x₁2+ y₁2+ z₁2= R2 and x₂2+ y₂2+ z₂2= R2 so
x₁2+ y₁2+ z₁2=x₂2+ y₂2+ z₂2.}

 

[chocolatelover]

Thank you very much

Regards