Understanding Subset Requirements in R2 and R3

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Discussion Overview

The discussion revolves around the relationship between R2 and R3, specifically addressing whether R2 can be considered a subset of R3. Participants explore the definitions and requirements for subsets, as well as specific examples, such as the plane defined by (x,y,0) in R3.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question why R2 is not a subset of R3 and seek clarity on the requirements for a set to be considered a subset.
  • One participant suggests that the space defined by (x,y,0) is a subset of R3, indicating a specific example of a plane within R3.
  • Another participant defines a subset as a condition where every member of one set must also be a member of another, arguing that since R2 consists of ordered pairs and R3 consists of ordered triples, no member of R2 can exist in R3.
  • A different viewpoint states that R2 is neither a subspace nor a subset of R3, emphasizing that two-component vectors from R2 cannot be derived from three-component vectors in R3.

Areas of Agreement / Disagreement

Participants express differing views on whether R2 can be considered a subset of R3, with some supporting the idea of specific planes being subsets while others argue against the notion based on the definitions of the sets involved.

Contextual Notes

Participants rely on varying interpretations of subset definitions and the dimensionality of vectors, which may lead to different conclusions about the relationships between R2 and R3.

elisemc
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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?
 
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elisemc said:
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.
 
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.)
 
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).
 

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