Expert: Paul Klarreich - 1/8/2010

Question
Dear Sir,
I have found the ollowing questions in my algebra book, which I did't understand well. Please clarify and answer them.

1. Every non-empty set has least member. This is also known as Well-Ordering principal. I understand the statement mentally. But, I dint find proof for this statement in my text book. Can you give a proof for it?

2. In the set of non-negative integers, zero is the smallest integer. How? because as per me, non-negative integer = positive integers = 1,2,3,... In this smallest element is 1. Where zero came? I dint undrestand. Morover, Zero is nether negative nor positive. How we can say 1 is least member in non-negative integers? please explain.

3.we know that |-4| = |4|= 4. But I have seen the following in my text book. |n| = n if n is greater or equal to 0. It is okay. I understand well. Also given, |n| = -n for n less than 0. How? because n<0 means -4. If we plug in mod, |-4| = 4 but not -4. Please explain this confusion. please...please...please...

with love,
mahima.

Answer
Questioner: mahima
Country: India
Category: Advanced Math
Private: No
Subject: basic
Question: Dear Sir,
I have found the following questions in my algebra book, which I did't understand well. Please clarify and answer them.

1. Every non-empty set has least member. This is also known as Well-Ordering principle. I understand the statement mentally. But, I didn't find proof for this statement in my text book. Can you give a proof for it?

>> No. It is a definition, not a theorem.

.................................
2. In the set of non-negative integers, zero is the smallest integer. How? because as per me, non-negative integer = positive integers = 1,2,3,...

>> Well, then. 'me' is wrong.  Non-negative means 0,1,2,3,...

'Non-negative' means just that. So zero is non-negative because, er... zero is not negative.

1,2,3,4.... means positive.  That is not the same.

..........................

In this smallest element is 1. Where zero came?

I dint undrestand. Morover, Zero is nether negative nor positive. How we can say 1 is least member in non-negative integers? please explain.

>> We don't say that.

....................................
3. we know that |-4| = |4| = 4.

Good.

But I have seen the following in my text book. |n| = n if n is greater or equal to 0. It is okay. I understand well. Also given, |n| = -n for n less than 0. How? because n < 0 means -4. If we plug in mod, |-4| = 4 but not -4. Please explain this confusion. please...please...please...

>> It has to do with the meaning of symbols.  When you write '-' it has no meaning all by itself -- it has to PRECEDE something.

When you write '-' its meaning depends on what comes after it:

'-' followed by a constant says 'negative'.

When you see '-4', what comes out of your mouth is 'negative 4'.

'-' followed by a variable or ()  says 'the opposite of'

When you see '-n', what comes out of your mouth is 'the opposite of n'.


So when you are writing

|n| = -n  for n < 0,

you say:

The absolute value of n is the opposite of n when n is negative.

Example:

|-4| = 4 because:

-4 is 'negative 4' and -4 is negative.  So,

The absolute value of -4 is the opposite of -4 because -4 is negative.

And the opposite of -4 is 4.