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June 3

Function question

Hello, this should be a relatively simple question. Recently I've learned about multi-variable functions, e.g. $f\colon \mathbb {R} ^{n}\to \mathbb {R}$ ; and of course I am familiar with single-variable functions, e.g. $f\colon \mathbb {R} \to \mathbb {R}$ . I realized that in both cases, no matter how many real numbers are input, the function only outputs a single real number. This got me wondering, can the following functions (or if not, perhaps equations instead) exist?

$f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}$
$f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n+1}$
$f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n-1}$

Thanks! 74.15.5.167 (talk) 02:29, 3 June 2016 (UTC)[reply]

Sure. You can have a function from any set to any set, except that the set on the right can't be the empty set, unless the set on the left is also the empty set. (If you allow partial functions, you don't even have that restriction.) --Trovatore (talk) 02:36, 3 June 2016 (UTC)[reply]

For an example of a function from Rⁿ to Rⁿ, see Matrix difference equation#Non-homogeneous first-order matrix difference equations and the steady state. For an example from R^n×n to R^n×n, see Matrix difference equation#Nonlinear matrix difference equations: Riccati equations. Loraof (talk) 03:05, 3 June 2016 (UTC)[reply]

Vector fields in Euclidean space are a common example of functions from

\mathbb {R} ^{n}

to itself, for instance. I'd think that you'll study the calculus of such functions once you study the calculus of real-valued functions on multivariate domains.--Jasper Deng (talk) 09:24, 3 June 2016 (UTC)[reply]

Point to clarify, regarding "single number". For a function F:R^n->R^n, the input and the output are both single elements of their respective sets. So if F(a,b,c)=(x,y,z), it is still sending a single input to a single output, even though we can characterize the output as an ordered n-tuple that sort of looks like a list of many numbers. For functions that sort of drop dimensions look at Projection_(set_theory) and projection_(mathematics). For A:R^n->R^m, consider

A\in \mathbb {R} ^{m\cdot n},\;x\in \mathbb {R} ^{n}

. Then using matrix multiplication, Ax=b is a mapping that can increase or decrease the dimension. SemanticMantis (talk) 14:42, 3 June 2016 (UTC)[reply]

A paper map of the Earth surface is a function from (some subset of)

\mathbb {R} ^{2}

, representing some range of (φ,λ), into (some other subset of)

\mathbb {R} ^{2}

, corresponding to (x,y) coordinates on a sheet of paper.
You can paint a picture of a landscape, which is a kind of a projection from

\mathbb {R} ^{3}

to

\mathbb {R} ^{2}

.
You can make a 3D model of the ocean bottom's depth, thus creating a real representation of

\mathbb {R} ^{2}\to \mathbb {R}

function.
You can describe your bike ride in time as

\mathbb {R} \ni t\mapsto (x,y)\in \mathbb {R} ^{2}

or a projectile trajectory in time as

\mathbb {R} \ni t\mapsto (x,y,z)\in \mathbb {R} ^{3}

Etc, etc, etc... --CiaPan (talk) 16:48, 3 June 2016 (UTC)[reply]

Symbol: X-like, but with a bar on top?

I believe once to have seen a math symbol looking like an X, the upper part being closed with a bar. Is there such a symbol, and if yes, what is it used for? --KnightMove (talk) 06:28, 3 June 2016 (UTC)[reply]

In probability theory

{\overline {X}}

usually means a mean value of a random variable

X

. --CiaPan (talk) 06:42, 3 June 2016 (UTC)[reply]

But that bar isn't closing the top of the X.

Looking over the "Mathematical Symbols" sections of the Unicode code charts, I can find symbols looking like an x with a bar closing any of the other three sides, but not the top! The three I found are charted on these two pages and identified as:

22C9 (⋉) left normal factor semidirect product
22CA (⋊) right normal factor semidirect product
2A32 (⨲) semidirect product with bottom closed

Note that the words "with bottom closed" seem to be describing the symbol rather than explaining its meaning. --69.159.60.83 (talk) 06:50, 3 June 2016 (UTC)[reply]

According to The Comprehensive LaTeX Symbol List, three different LaTeX symbol packages have the top-closed product as \utimes, suggesting that someone has used it for something, but I don't know who or what. -- BenRG (talk) 08:36, 3 June 2016 (UTC)[reply]

(ec) In a set theory it is a complement of a set, and in Boolean algebra it is sometimes used for negation. --CiaPan (talk) 06:42, 3 June 2016 (UTC)[reply]