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).

 

[blue_raver22]

I can provide a response to the concept of subset requirements in mathematics. In order for something to be considered a subset of another set, it must fulfill the condition of being "contained in" that set. This means that every element of the subset must also be an element of the larger set.

In the context of R2 and R3, R2 represents a two-dimensional space while R3 represents a three-dimensional space. Therefore, R2 is not a subset of R3 because it does not fulfill the condition of being "contained in" R3. R2 and R3 have different dimensions and therefore, their elements cannot be interchangeable.

To be considered a subset, the space (x, y, 0) would need to fulfill the condition of being "contained in" R3. In this case, the third dimension (z) is fixed at 0, making it a two-dimensional space. Therefore, (x, y, 0) can be considered a subset of R3 as every element of (x, y, 0) is also an element of R3.

In conclusion, for something to be considered a subset, it must fulfill the condition of being "contained in" the larger set. In the case of R2 and R3, their different dimensions prevent R2 from being a subset of R3. However, the space (x, y, 0) can be considered a subset of R3 as it fulfills the condition of being "contained in" R3.