confused between two sets

[logistic_guy]

a. the set of positive rational numbers

b. the set of nonnegative rational numbers

why they're different?

 

[R.M.]

Can you think of a number that belongs to the second set but not the first? HINT: think about the difference between "positive" and "nonnegative".

 

[Dr.Peterson]

a. the set of positive rational numbers

b. the set of nonnegative rational numbers

why they're different?

What country are you from, and what language do you speak? I recently became aware that in at least one language, "positive" is taken to include zero, which is not true in English. For example:

Negative number - Wikipedia

en.wikipedia.org

The convention that zero is neither positive nor negative is not universal. For example, in the French convention, zero is considered to be both positive and negative. The French words positif and négatif mean the same as English "positive or zero" and "negative or zero" respectively.​

In such a language, it is understandable that you would be confused, because "positive" and "non-negative" would mean the same thing. If you are a native English speaker, then you should not be; "positive" means "greater than zero", while "non-negative" means "not less than zero", and therefore includes zero.

And in any case, we need some way to distinguish "numbers greater than zero" from "numbers greater than or equal to zero".

 

[logistic_guy]

Arabic. Bahrain. i thought positive and nonnegative means the same thing

i understand now set a have elements, say, \(\displaystyle k\), \(\displaystyle k > 0\) where \(\displaystyle k\) can be expressed as \(\displaystyle k = \frac{A}{B}\) where \(\displaystyle A\) and \(\displaystyle B\) are integers. set b have \(\displaystyle k \geq 0\).

including zero in set b is a contaradiction as both sets are not well-ordered.

proof: qoute this from google A nonempty set of positive integers is said to satisfy the well-ordering property if it has the least element. set b is nonempty and have the smallest element zero, so it is well ordered. isn't this a contadiction?

 

[Dr.Peterson]

including zero in set b is a contradiction as both sets are not well-ordered.

proof: quote this from google A nonempty set of positive integers is said to satisfy the well-ordering property if it has the least element. set b is nonempty and have the smallest element zero, so it is well ordered. isn't this a contradiction?

What are you saying contradicts what? Your question was about the distinction of the two sets, and made no mention of well-ordering. Both sets obviously do exist; they just have different properties. Moreover, the two sets you ask about are sets of rational numbers, not of integers! So what are you talking about?

What you claim to quote "from Google" (Google is not itself a source, but a way to find sources, which may not be trustworthy), while very poorly stated, only mentions sets of positive integers, so it doesn't say anything about the set of non-negative integers, or of positive rational numbers. That makes it irrelevant to your claim, even if it were correct.

Wikipedia says this about the positive integers:

In mathematics, the well-ordering principle states that every non-empty subset of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which x precedes y if and only if y is either x or the sum of x and some positive integer (other orderings include the ordering 2 , 4 , 6 , . . . ; and 1 , 3 , 5 , . . . ).​

But the general definition of a well-ordered set doesn't apply only to positive integers:

In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set.​

None of this is relevant to your question.

 

[logistic_guy]

you're right Dr.Peterson, the well ordering properti i give is not applicable in a set conatain rational numbers.

i'll look at

Wikipedia says this about the positive integers:

In mathematics, the well-ordering principle states that every non-empty subset of positive integers contains a least element. In​

i'll try to prove set b isn't well-ordered even with zero as element of it. but i think that i've to search for a smilar property for

non-empty subset of positive rational numbers​

if not exist, i'll try a contradiction

thank you Dr.Peterson. at least now i understand the difference between set a and set b.