Understanding Subset Requirements in R2 and R3

[elisemc]

Why is R2 not a subset of R3? And then, what are the requirements for something to be a subset? I vaguely understanding that it has to be "contained in"

Would the space (x,y,0) be a subset of R3?

 

[mathman]

Why is R2 not a subset of R3? And then, what are the requirements for something to be a subset? I vaguely understanding that it has to be "contained in"

Would the space (x,y,0) be a subset of R3?

R2 not a subset of R3 - could you be precise?

A particular plane (x,y,0) for all x and y, is a subset.

 

[HallsofIvy]

Set A is a subset of set B if and only if every member of A is a member of B.
R² consists of all ordered pairs of numbers, (x, y). R³ consists of all ordered triples of numbers, (x, y, z). A pair is not a triple so no member of R² is in R³.

(We can associate the pair (x, y) with the triple (x, y, 0), for example so that R² is isomorphic to a subset of R³.)

 

[h6ss]

ℝ² is neither a subspace or subset of ℝ³ because any two-component vector from ℝ² cannot come from a set of three-component vectors, in particular ℝ³. In other words, the vector (a,b) is not the same as the vector (a,b,0).