30
votes
Accepted
Math Proofs - why are they important and how are they useful?
Proofs are important because proofs are just understanding how we know that something is true. This is what mathematics is all about!
What if all you care about is using the results of mathematics: ...
- 22.9k
27
votes
Accepted
What's the point of learning equivalence relations?
Perhaps emphasize to students the spirit of equivalence relations. They partitions sets into equivalence classes--cutting down the amount of cases necessary to prove something.
To illustrate this, ...
- 495
20
votes
Math Proofs - why are they important and how are they useful?
I am an engineer. I have not done a mathematical proof since leaving school. Despite that, I believe that proofs are the second most important skill that any student will learn during their entire ...
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18
votes
Math Proofs - why are they important and how are they useful?
I am an engineer.
Proofs are important to "get" engineering, but are not directly used. I see three aspects of learning proofs as important: Logic, Process, and Ontology.
Logic is the ...
- 280
16
votes
What's the point of learning equivalence relations?
In general, all applications are going to be more of the "here is what we have to check to make sure that our algorithm/theory/definitions work". We don't usually encounter practical ...
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15
votes
Accepted
Proofs that make theorems less clear
"A well-chosen example illustrates ... and is entirely convincing."
For me, all of what is usually called "generic proof" satisfy your criterion. Consider the Euclidean algorithm for finding the ...
- 4,332
11
votes
What's the point of learning equivalence relations?
Equality vs. Identity
Alexei touched on this topic by mentioning hash tables, but I would like to spell it out more explicitly, because this is a critical and fundamental topic in software engineering,...
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11
votes
Math Proofs - why are they important and how are they useful?
Along similar lines as the previous answers. Just as a computer program needs to be tested, to make sure it works correctly, a mathematical result needs to be proved to make sure it really does work.
...
- 18.9k
10
votes
Is Calculus Necessary?
Society always benefits when new mathematical tools are developed to get insights into the world around us. There is little doubt that Calculus has helped change the quality of the lives we can lead ...
- 1,862
9
votes
Counterexamples to the Greedy Algorithm
The greedy algorithm fails quite spectacularly for the Traveling Salesman Problem (TSP):
Bang-Jensen, Jørgen, Gregory Gutin, and Anders Yeo. "When the greedy algorithm fails." Discrete Optimization 1....
- 28.7k
9
votes
Proofs that make theorems less clear
There are many situations in which we have a clear collective understanding of intention, or goals, and examples which persuade us that these goals are plausible, as well as illustrating apparent ...
- 13.8k
9
votes
Accepted
Is playing and teaching chess appropriate in private lessons?
It's a bad idea. (1) It's not that special. He can get a game from chess very easily from anyone. An individual session with you is not high value, not best use of time. (2) It will raise hackles.
...
- 116
9
votes
Simple combinatorics problems using division
You can do necklaces composed of colored beads. For simplicity the number of beads needs to be prime. If there are to be $p$ beads with $a$ choices of color, then there are $a^p$ ways to arrange the ...
- 1,088
9
votes
Math Proofs - why are they important and how are they useful?
As in @StevenGubkin's answer, indeed, "proofs" are (fairly definitive) explanations why something is true.
I would agree/concede that mathematics is very useful to a variety of people ...
- 13.8k
7
votes
Accepted
Teaching logic through "high school algebra"?
Obviously, one place to look is in the huge amount of “new math” curriculum material that was written during the late 1950s to early 1970s, but I’ll leave that for you or someone else to search ...
- 5,542
7
votes
Textbook to study group theory as a part of Discrete Mathematics
The list of topics you want to study corresponds rather to abstract algebra than group theory. You did not say why you are interested specifically in group theory, but I believe that acquaintance with ...
- 269
6
votes
Counterexamples to the Greedy Algorithm
There are tons of tasks where greedy algorithms fail, but the best in my opinion is the change-making problem. It is great, because whether the obvious greedy algorithm works depends on the input (i.e....
- 8,807
6
votes
What's the point of learning equivalence relations?
If you have looked at modular arithmetic, then one possibility is: Give/recall some example of an algebraic argument in modular arithmetic; then point out that the argument is implicitly relying on ...
6
votes
What's the point of learning equivalence relations?
The notion of equivalence relation is one of the basic building blocks out of which all mathematical thought is constructed. (Paul Halmos)
What's the point of learning equivalence relations?
The ...
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6
votes
Mnemonics to correlate the definition of "asymmetric relation" and "antisymmetric relation" with the terms
First, let's note that the terms as used by Rosen are standard definitions, as we can see on Wikipedia (here and here), as well as other resource sites. There was some question about this in the ...
- 21.5k
6
votes
What's a good notation to show elements of relation composition?
To explain why any particular ordered pair is in $S \circ R$, you can just show that it satisfies the definition, which says that $(a,c)$ is in $S \circ R$ if there exists $b$ such that $(a,b) \in R$ ...
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6
votes
What's a good notation to show elements of relation composition?
I'd avoid introducing a new notation for such a limited scope: you won't probably use it elsewhere during the class, and it wouldn't be used elsewhere in the literature.
I have three suggestions. The ...
- 498
6
votes
Math Proofs - why are they important and how are they useful?
You need a proof to know that what you've observed is true beyond the cases where you observed it.
How do you know that the sum of two odd numbers is even? Most kids say it's obvious after they've ...
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5
votes
Textbook to study group theory as a part of Discrete Mathematics
To repeat my comments:
I can recommend Fraleigh's classic introduction.
It is easy to read for beginners, with many exercises, from easy to difficult, on which self-learners could check themselves.
...
- 28.7k
5
votes
Accepted
Tutoring Discrete Mathematics
I have found the best thing to do in tutoring a proof based class is to constantly ask the questions "why is that true?" or "what is the definition of that?". These questions help them build relations ...
- 2,594
5
votes
What's the point of learning equivalence relations?
Dictionary/hash table relies on equivalence to bucketize items.
So knowing that one would never try to build a hashtable by a distance between cities (objects on a plane): distance is not transitive.
...
- 151
5
votes
College undergraduate geometry courses
I taught a geometry course which counted both for math majors (sophomore level), and was required for the math education program secondary education teacher majors, and we used the text:
Edwin Moise: ...
- 2,065
5
votes
Accepted
History of discrete math curriculum
This is not a question that can be easily answered without careful historical research. Here are some guesses in the absence of such research.
What's been added?
Algorithm analysis: big-$O$ notation. ...
- 28.7k
5
votes
Math Proofs - why are they important and how are they useful?
Proofs are the whole point of mathematics. They are how we verify and explain that we know things instead of merely guess at them. When I personally teach discrete mathematics, the first-day opening ...
- 21.5k
4
votes
Is Calculus Necessary?
I don't believe that there is much of debate about Calculus today. Major universities continue to push the overly rigorous vision embodied in texts like Stewart or Hughes Hallet. The AB/BC Calculus ...
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Only top scored, non community-wiki answers of a minimum length are eligible