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Questions tagged [definition]

For requesting, clarifying, and comparing definitions of mathematical terms.

Definitions are at the core of mathematical precision; they answer the question "What is X?" in mathematics. Into this category fit questions regarding equivalence of definitions, clarification of complicated definitions, or proposed new definitions for mathematical notions, with requests for improvements or comments.

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Does this make sense?

I'm trying to express a function greater than zero defined on countable sets dense in $\mathbb{R}$ that are the quotient of countable sets dense nowhere. Consider $P:A\to\mathbb{R}$ where $P(x)> 0 \text{,} \ $ $\mu(A)=0$ and $$P(x)=…
Arbuja
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How to describe a piece-wise function in words?

Suppose we want to describe a peice-wise function in words. My last post was down-voted because it was not clear. Here is what I'm trying to desribe. Function $F:D\to\mathbb{R}$, $D\subseteq \mathbb{R}$, $\bigcup_{i=1}^{\infty}A_n=D$…
Arbuja
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How to find Df in functions

Well, I do understand what Df is and how you find it in simple equations, however, I am kinda confused in "complex" functions. For example, the following functions: 1* f(x)=x^3+x^2-x-1 , Df=R (however, I don't understand why.) 2* f(x)=2x/(1+x^2), we…
user2041143
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Definition of "of odd order"

Could someone point me to a definition of "odd order"? The definition of group order I found here seems to refer to a group as a set of numbers. The context that I'm reading about 'odd order' is: "$p\ \alpha\ d_{eh_1}E(t)$; it is initially linear…
mattzhu
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"Monotonically increasing integers" exact definition

If nothing else is specified, does the term "monotonically increasing integers" mean that $$f(k+1)+1 = f(k), \forall k \in \mathbb{N}$$ or $$f(k+1) \geq f(k), \forall k \in \mathbb{N}$$
Alex5207
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A simple question about uniqueness terminology

I am reading through a book on topology and came across the following phrase: for each element $x\in X$ we can uniquely define a function $f_x: A\rightarrow B$ such that $f_x$ satisfies some topological property $P$. What exactly does this mean?…
fosho
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Is defining $\mathbb{R}$ as $\{e \mid e^2 \ge 0\}$ good for non-mathematics undergraduates?

Since the question might be unclear, let me provide some details about how I came to asking it: Providing rigorous definitions to non-mathematics undergrads is a non-trivial task -- yet important. However, the equilibrium between rigour and…
DeeCeeDelux
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What is the mathematical definition of "minimum" when describing a real number?

In mathematics, is the word "minimum", when used to describe a real number, conventionally taken with respect to magnitude only or magnitude and sign? For example, a question asks for the "minimum displacement" of a particle over a certain time…
PKBeam
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Why do mathematicians define the straight angle to be $180^\circ$?

The answer for How do you know a straight line forms an angle of 180°? is (by users in the post) that A straight line, by definition, is $180^\circ$. I wonder why mathematicians don't chose any other number but $180$ for the definition of a straight…
Akira
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Using if and not "if and only if" in definitions

We often see the word "if" in connection with definitions e.g y is a solution of $x^{'}=f(x,t)$ if $y^{'}=f(y,t)$ or A metric space is complete if every cauchy sequence converges. Why isnt it written if and only if in general in this context?
Bora Bora
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Priority of logarithm

In an expression like the following one, which one of log or multiply does have priority? In other words, does it equal $\log(V\times V)$ or does it equal $\log(V)\times V$? $\log V \times V$
Shayan
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What's the relationship of this definition and the injective function?

I have the following definition: Let $f$ be a function. We say that $f$ is injective if $(a, y)\in f$ and $(b,y)\in f$ (i.e., $f(a) = f(b)$) then $a = b$. I understand the last sentence but I cannot establish the relationship between the…
Matheus
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What helps or what does this restriction mean in the statement

I have done quite a few exercises, and I have never considered some restrictions, for example: For m $\neq 0$ , $\dfrac{m+n}{m} - \dfrac{n-m}{m}$ Resolving: = $\dfrac{m + n - n + m}{m}$ = $\dfrac{2m}{m} = 2$ Well, the answer is correct. I even…
ESCM
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Definition of a non-linear function

I would like to use in a proof that $f(x)$ is a non-linear function of $x \in \mathbb{R}$, without assuming that $f$ be differentiable/analytic. I can also use that $f(0)=0$. Q: What is a convenient definition for this? If $f$ were differentiable,…
mts
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For any $x>0$, does $|a-b| < x$ imply that $|a-b|=0$?

Let $a\in \mathbb{R}$. For any $x > 0$: $$ |a-b| < x $$ implies that: $$ a=b $$ Is this statement true? Why? My attempt: I think it is true. My reasoning is that, if $x$ is allowed to be anything larger than $0$, but not $0$, it will contain an…
caveman
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