# Questions tagged [definitions]

For questions related to the issue of concepts of definitions.

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### Define logarithmic function by functional relation [closed]

My son was working the other day with exercises such as: Find all the mappings $f:\mathbb{N}\rightarrow\mathbb{Z}$ verifying $$\forall m,n \in \mathbb{N}, f(m+n)=f(n)+f(m).$$ As another example: Find ...
• 165
1 vote
302 views

### Is there a particular reason why segment addition postulate and partition postulate are two different things?

I could be wrong but those two ideas sound the same, just that the partition postulate is more general. There is also the angle addition postulate. The segment addition postulate states that if three ...
• 1,068
1 vote
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### Is it correct to state that a cone has no faces?

Faces are attributes of polyhedra, so it doesn't make sense to ask how many faces a cone has. Are there traditional scholars that use faces attached to cones? How do different countries deal with the ...
1 vote
324 views

### Are there differences between graphs, diagrams and charts?

"Can you explain the distinctions between graphs, diagrams, and charts, and provide definitions for each of these concepts? Specifically, is every graph considered a diagram? Are graphs ...
2k views

### Can this be a better way of defining subsets?

I remember my high school days where subsets were defined in the following manner: Given two sets A and B, if every element of B is an element of A, then B is called a subset of A. A common ...
157 views

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### Why is continuity defined as a local property?

The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the ...
• 2,031
2k views

### How can I motivate the formal definition of continuity?

In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition $f$ is continuous when its graph can be drawn without lifting the pen. ...
• 2,031
605 views

### Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
2k views

### What is the intuition behind the limit superior?

I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
• 2,031
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### Rigorously defining the concept of an angle for high school students

Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected. Google search provided me with many definition that are too technical or too vague IMO. ...
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### Can students tell the difference between the "definition if" and the "theorem if"?

The word "if" is used in two meanings in mathematics: Definition. A topological space is compact if every open cover has a finite subcover. Theorem. A topological space is compact if it is ...
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### What are the questions one should ask of oneself in trying to understand definitions and lemmas? [closed]

Definition (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then ...
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### Why is distribution prioritized over combining?

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [\$a(b + ...
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### Is multiplication by zero clear for and understood by K-3 students?

For K-3 students, perhaps it is not acceptable to introduce multiplication by zero as a property or definition. Instead, the child may think about multiplication as, e.g., repeated addition. Examples ...
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### how is volume different than capacity

I have in mind that volume is the amount of room or space a 3-d object takes up - its "outsideness" and that capacity with the amount of room or space a 3-d object can hold. Then I start thinking ...
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