Wednesday, July 21, 2010

Is the question stupid?

Questions which dont make sense.

Question 1. Which of the following columns is greater?

Column A   The number of integersthat are less than -10
Column B   The number of integers that are greater than 9

The answer they provide goes something like this.

First, you'll need to know that there are as many negative integers as there are positive integers, because for every positive integer,

there is an equivalent negative integer. However, the set of numbers in Column B is greater than the set of numbers in Column A because

Column B also includes positive 10, while Column A does not include negative 10.

I have a counter argument for this.

For the time being, lets map -11 to +10.
Now I would have to map -12 to +11.
-13 to +12 etc.
If I represent -11 as x and +10 as y (x and y being constants), the mapping becomes like this.

X -> Y
x-1 -> y+1
x-2 -> Y+2.

Now I denote the numbers added to x and y as z (z is a variable which increases by one always).

X-Z -> Y+Z . Note that Z is the same for both L.H.S and R.H.S.

The mapping holds true irrespective of the value of Z, even when  z-> infinity ( basically we say that Z tends to infinity). . For any value of Z from 1 to infinity, there is a number X-Z which can map to X+Z.

Hence by the law of induction, both are equal.

Column A is infinity and column B is infinity. This is the reason we cannot compare one infinity to another.

Do let me know your views. :)

Joseph said...

Your argument would be stronger if you defined your mappings as bijections to the natural numbers {0,1,2,3...} which would make both sets countably infinite and therefore have the same cardinality. The sets would be considered the same size.

It is easier to get than to keep it.............................................................

bokk said...

Never put off till tomorrow what may be done today.......................................................................