How can i resolve this.
let w = {(x, y, z) |x^2+y^2+z^2| ∈ r3}. prove that w is a subspace of r3.

0 + 0 + 0 ≠ 1.

Step-by-step explanation:

Additive identity is not in the set so no subspace.

guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation: