Can any subset of <mi mathvariant="double-struck">R 2 </msup> can be expressed by e

ownerweneuf

ownerweneuf

Answered question

2022-05-20

Can any subset of R 2 can be expressed by equations /inqualities?
For example the set of all points on a circle in R 2 can be expressed by an equation. Similarly square, rectangle, parabola, interior of a circle, triangular regions, etc.
Likewise, can any subset of R 2 can be expressed by a system (finite or infinite number) of equations or by inequalities?

Answer & Explanation

Michaela Alvarado

Michaela Alvarado

Beginner2022-05-21Added 11 answers

Sure. Take the indicator function of the subset = 1. Then, the subset is the set of points such that that equation is true.
Jaycee Mathis

Jaycee Mathis

Beginner2022-05-22Added 6 answers

If you only allow finite many formulas (or equivalently, a single formula by concatenating all of them), then, interpreting your question as "Can every subset of R 2 be defined by a formula in which you allow reals as parameters?"
then it is one of those 'typical' set-theoretic questions which can't be answered using our axioms of set theory¹. We cannot prove that there exists such an "undefinable" set, and we cannot prove that no such set exists.

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