Real Number Sets: Notation Explained | Additive Expressions

[QuestForInsight]

Hello, everyone. I've trouble attempting to read the following. One of the things we assume for the set of real numbers is.

• A map $\left(\xi, \eta\right) \to \xi+\eta$ from $\mathbb{R} \times \mathbb{R}$ into $\mathbb{R}.$

Could someone read the above in plain English, please. Does it mean all ordered pairs in $\mathbb{R}$ can be expressed additively?

 

[Chris L T521]

Hello, everyone. I've trouble attempting to read the following. One of the things we assume for the set of real numbers is.

• A map $\left(\xi, \eta\right) \to \xi+\eta$ from $\mathbb{R} \times \mathbb{R}$ into $\mathbb{R}.$

Could someone read the above in plain English, please. Does it mean all ordered pairs in $\mathbb{R}$ can be expressed additively?

Almost - it means that any ordered pair in $\mathbb{R}\times\mathbb{R}$ can be related to a real number by adding together the components of that ordered pair. In other words, we have a map $f : \mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$, where $f(\xi,\eta) = \xi + \eta$.

Aside: Note that if we had another map, say $g:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$, $g(\xi,\eta)$ can be something completely different; for instance, we could have $g(\xi,\eta)=\xi\eta$ (the product) or $g(\xi,\eta) = \xi^{\eta}$ (exponential). How we define these maps depends on the context of the problem.

I hope this helps!

 

[QuestForInsight]

Thanks. That clears it up. I've another one, if you don't mind.

What does $(\xi, \eta) \to f(\xi, \eta)$ from $\mathbb{G} \times \mathbb{G}$ into $\mathbb{G}$ mean?

EDIT: Ignore it! I should have paid more attention to your post!

 

[QuestForInsight]

I think I get it. It means any ordered pair $(\xi, \eta)$ in $\mathbb{G}\times\mathbb{G}$ can be related to an alement in $\mathbb{G}$; the nature of this element depending on what operations we allow $f(\xi, \eta)$ to obey -- addition, multiplication, exponentiation etc.

 

[Chris L T521]

I think I get it. It means any ordered pair $(\xi, \eta)$ in $\mathbb{G}\times\mathbb{G}$ can be related to an alement in $\mathbb{G}$; the nature of this element depending on what operations we allow $f(\xi, \eta)$ to obey -- addition, multiplication, exponentiation etc.

Well, more generally, we can define $f(\xi,\eta)$ to be any function dependent on $\xi$ and $\eta$ (as long as it is defined on $\mathbb{R}$, I believe) that outputs a real number. For instance, we can take $f(\xi,\eta)$ to be much more complicated things, say like $f(\xi,\eta) = \xi\eta-\cos(\xi-\eta) + \sinh\eta$. But it all boils down to what's going on in the problem at hand. I would assume with where you're at, it would be a little more basic, defining the values in terms of addition, multiplication, exponentiation, etc.

 

[Evgeny.Makarov]

A map $\left(\xi, \eta\right) \to \xi+\eta$ from $\mathbb{R} \times \mathbb{R}$ into $\mathbb{R}.$

A small remark about notation. When one writes the type of a function, i.e., its domain and codomain, one usually uses an arrow: e.g., $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$. When one writes the definition of a function, i.e., the rule that maps an argument into a result, one usually uses $\mapsto$, e.g., $(\xi, \eta) \mapsto \xi+\eta$. The LaTeX command for $\mapsto$ is in fact \mapsto. See Wikipedia.

 

[QuestForInsight]

My book uses just the arrow and the one that starts with the bar doesn't appear in it at all.

It was written in 1969, so maybe that explains it (or perhaps not). Thanks for letting me know.

 

[CaptainBlack]

My book uses just the arrow and the one that starts with the bar doesn't appear in it at all.

It was written in 1969, so maybe that explains it (or perhaps not). Thanks for letting me know.

Which book is it?

CB

 

[QuestForInsight]

Which book is it?

CB

Linear Algebra and Geometry - Jean Dieudonné.

 

[CaptainBlack]

Linear Algebra and Geometry - Jean Dieudonné.

Unless you have had some prior exposure to the topics covered I would not use this book (and certainly not if I wanted to learn about geometry).

Dieudonné is/was a leading light of the Bourbaki group who were dyed in the wool formalists, so unless you are looking for a formalist presentation you should avoid their works.

(Formalism in this sense is the school of mathematics that regards mathematics as a game played with symbols following a set of rules, devoid of any meaning. This goes back to Hilbert who hoped to reconstruct all of mathematics this way, the Hilbert program was essentially invalidated/derailed by Gödel's and Turing's work in the 1930's. Bourbaki aimed to use set theory as the starting point for the formal reconstruction of mathemetics, but this is still subject to the Gödel limitations).CB

 

[QuestForInsight]

Thanks for the advice. I'll stick with it for a bit, and if I think I'm not getting anywhere or it becomes too much, I'll leave it. As you have probably gathered, I was trying to construct the definition of Abelian group using his notation in that other thread you were helping me with earlier.