Expert: Josh - 1/8/2010

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 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
Dear Mahima,

1. If we are dealing with a finite set S that contains |S|=N elements, we can use the "bubble sort" procedure to order each element of the set in ascending order. [See Wikipedia (http://en.wikipedia.org/wiki/Bubble_sort) for an example on how this works]

Following this procedure, we identify the first element as the smallest.

2. The set of non-negative integers {0,1,2,...} actually contains zero as well as all the positive integers, "0" belongs simply because it is not a negative number. On the other hand, the number "1" is the smallest element in the set of positive numbers {1,2,...}. Non-negative is NOT the same as positive. By the same token, non-positive is NOT the same as negative. The set of non-positive integers is given by {...,-2,-1,0}.

3. When n is negative, -n is a positive number. Consider n=-4, for instance, its absolute value is |n| = -n = 4.