Discrete Math-Cartesian Product Question

[discretec]

Instructions: List the elements in the set.
Problem: A={(a,b)belongs to NxN such that a is less than or equal to b, b is less than or equal to 3}
N being natural numbers.
For my answer I have this:
A={(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}
The problem I am having is I am not sure if the answer should be in ordered pairs as I have put here. I know the Cartesian product is for combining 2 different sets. a,b are both in the same set A in this problem. So, my question is are the ordered pairs correct or should I actually multiply each ordered pair?
Thanks in advance for any help.
C

 

[pka]

Instructions: List the elements in the set.
Problem: A={(a,b)belongs to NxN such that a is less than or equal to b, b is less than or equal to 3}
N being natural numbers.
For my answer I have this:
A={(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}

This answer is correct if \(\displaystyle 0\not\in N\).
Some textbooks have \(\displaystyle 0\in N\).
So you need to check your text material and possibly adjust.

 

[discretec]

Thank you. I appreciate you looking at my problem.
C

 

[mmm4444bot]

Some textbooks have \(\displaystyle 0\in N\).

That's sloppy, if not distressing.

 

[pka]

Re:

Some textbooks have \(\displaystyle 0\in N\).

That's sloppy, if not distressing.

Actually it is neither sloppy nor distressing if one knows something of the history.
http://mathworld.wolfram.com/NaturalNumber.html
http://en.wikipedia.org/wiki/Natural_number

 

[mmm4444bot]

… if one knows something of the history …

The history of the number zero?

I know some things about the history of the number zero; so, why do I nevertheless feel distressed? :wink:

(I've read "Zero: the biography of a dangerous idea" by Seife.)

Thank you, PKA, for the reference to Ribenboim's 1996 statement that we can make zero a Natural number "whenever [it is] convenient". I do enjoy laughing out loud.

I'm willing to concede that my unsolicited opinion in this thread might be misplaced regarding "Advanced" courses. What motivated my post herein? Apparently, I'm still sensitive over being belittled by a UW professor because I stated "the set of Whole numbers" (in which the professor does not believe) instead of "the set of non-negative integers".

We spend years drilling certain facts into the students' head, only to later change foundation(s) "whenever [it is] convenient".

What's wrong with calling the Naturals plus zero the set of Whole numbers? (This, too, is a rhetorical question.)

Oh well.

It's a free world, brother.

See ya 'round.

 

[daon]

I have a professor this year who said "it is nonsensical to think 0 is not a natural number." So yes, opinions do differ.

I still use terminology like "whole number" and see no problem with it. Someone who wants to make fun of ME for my adopted terms can go to h*ll. While I will adapt to the definitions used in a particular course, I wouldn't hold any professional respect for someone who would rather nitpick about the synonym I used than to hear my message. I commonly interchange "non negative integer," "natural number" and "whole number." If I would have a student who tells me zero is not a natural number, I would give them a little smile, say "sure," and continue the proof.

I don't disgree with either definition of N, and I will use both depending on the context without issue. It has changed in several classes for me, and I find that kind of humerous.

 

[mmm4444bot]

My concern is that a secondary or undergraduate student will come across such a post that defines zero as a Natural number right after it's been drilled into their head that zero is a Whole number.

I do not like seeing or hearing about student confusion directly resulting from inconsistent, sloppy, or contradictory information, at the "non-advanced" level.

That's all.

(Such students do still actually read posts at this site. Am I correct?)

 

[pka]

Re:

My concern is that a secondary or undergraduate student will come across such a post that defines zero as a Natural number right after it's been drilled into their head that zero is a Whole number.

I actually agree with that sentiment.
That why I do not think either word should be used.
We have two perfectly clear and unambiguous descriptions: positive integers or non-negative integers.

My thesis advisor told me, almost forty years ago, if you use an ambiguous term it will be misunderstood. He would allow any of his students use either term: natural number or whole number. (Of course, he also did not think a set could be empty).