How can I accurately read mathematical symbols to understand 3D space?

[Neek 007]

I am not too used to reading mathematical text using symbols. I need some help confirming what I have in my textbook.

The book is describing a point in 3D space using the definition

R x R x R = {(x,y,z) | x,y,z $\in$ R} and R is all real numbers


Using this source describing the definition of each symbol: http://en.wikipedia.org/wiki/List_of_mathematical_symbols

I came up with this definition in words

The Cartesian product of R and R and R is equal to the set consisting of (x,y,z) such that x,y,z are elements of R.

Is this an accurate definition in words? I'm finding that being able to read the precise definition from math symbols is becoming even more important as my courses continue. Any other input on reading math symbols is appreciated and encouraged.

 

[Antiderivative]

You're right in that you need to be able to learn what mathematical symbols mean as a way to determine shorthand. It's very important in set theory (and real analysis, which is a total pain but appreciable nevertheless).

Your definition states that "The Cartesian product of R, R, and R is equal to a set of three real numbers x, y, and z". That just describes 3D space really... 3D space is R^3, or the set of all real numbers in three dimensions. By setting x, y, and z separately equal to a real number and creating a vector space that consists of all possible linear combinations of x, y, and z, we span out all of 3D space (or R^3). Your book likes to use some technical jargon instead of linear algebra, which makes more sense intuitively from a non-mathematical standpoint.

 

[Neek 007]

Okay, thank you. The textbook goes on from their definition to say "We have given a one-to-one correspondence between points P in space and ordered triples (a,b,c) in R^3".

I think I understand this statement to be the Point P is located at (a,b,c). Is this also a fair comprehension?

 

[HallsofIvy]

Not quite, because it is not talking about a single point, P. What it is saying is that every point in 3 dimensional space can be associated with a triple of numbers.

 

[Antiderivative]

Indeed, HallsofIvy is correct. The book is talking about the set of all points in 3D space and how one can associate ordered triples of the form (a,b,c) with each such point. The book's statement also implies that these points (a,b,c) are unique in describing a single location in 3D space (hence a one-to-one correspondence).

 

[Neek 007]

Thank you! After reading a second time, I noticed it said "points", plural, and that lines up with the explanation. Thanks for helping me.