What is the meaning of equation of a circle or equation of a cube?

Question

We have an equation of a Circle $$x^2 + y^2 = 1$$ and equation of a Cube $${||x-y|+|x+y|-2z|+||x-y|+|x+y|+2z|=1}$$

When I substitute x,y in first equation or x,y,z in the second equation then the equations are said to be satisfied when LHS and RHS are equal.

I have a quick question. If the first equation or second equation get satisfied, because LHS becomes equal to RHS, then those values of x and y of first equation are said to be lying on the circumference of the circle and not inside or outside of the circle and x,y,z of the second equation are said to be lying on the surface of the cube and not on the inside and not on the circle of the cube. Correct?

Answer 1

Yes. In set-builder notation, one would say the following

The circle $S$ consists of of $$\{(x,y) \in \mathbb R^2 \mid x^2+y^2=1\}.$$

And same goes for the second one.

Answer 2

Yes, this is what those equations mean.

In general, a relation defined by an equation is graphed as the set of all points that satisfy the given equation. So exactly the same as you would graph a line, a parabola, or non-function relations like these two. In fact, disregarding the visual side of things, some people refer to the concept of the set of points satisfying the equations as the graph of the given object.

To find all the points inside the circle, you could use an inequality like

$$ x^2 + y^2 \leq 1 $$