Okay, all I need to know is how my book gets the equations to define sets when t

Question

Okay, all I need to know is how my book gets the equations to define sets when trying to prove cardinality. For example in this problem:

Prove that the following set has cardinality c:

(a, infinity) for any real number a

My book says: Hint: Let f(0,1)->be defined by f(x) = (a-1)+(1/x)

Where do they get this equation from? I feel like it was just pulled out of the air.

Same question for this one:

Prove that the following set has cardinality c:

(0, 1]U(2, 3]U(4, 5)

My book says to define the set by: f (x) =2 ? 2x if 0 < x ? 1/2
2x + 4 if 1/2 < x < 1

Same question: where do they get this equation?

Will rate extremely fast for clearly explained answers. Thank you!

Explanation / Answer

I assume you are aware of the concept of cardinality which is basically the number of elements of the set. To prove the cardinality of a set to be c which is a finite constant, we basically bijectively map the set to (0,1) using a bijective function f(x) for x in (0,1) since we know the cardinality of (0,1) is c.

Now we must understand that their may be several possible bijective f(x) which we can use for this purpose and we just need to think of and give one.

There is no formal method or formula to do this but we follow a simple approach which I explain using part a as an example.

Part a has a single interval namely (a, infinity) so we consider the interval (0,1) as a whole. In this approach we choose an increasing or decreasing f(x) with value of f(0) to be one end of given interval and value of f(1) the other end. Here, the simplest way to get infinity is at x=0 where 1/x will be infinite. So f(x) has 1/x as one term. Now for the other end, f(1) must be equal to a. But 1/x =1 at x=1. So we simply add (-1) and a to f(x) and both conditions get fulfilled. f(x)=1/x+a-1. Now we see that f(x) is continuously decreasing in (0,1) so all values in between (a,infinity) will be obtained. Hence chosen f(x) is sufficient.

By this approach, in part b, I would break (0,1) in 3 intervals as we are to prove cardinality of (0, 1]U(2, 3]U(4, 5) which has 3 intervals itself. These intervals may be equal for convenience and in a similar manner, I would map each of the 3 intervals of (0,1) to the given 3 intervals. I may obtain a different answer than given but it would nonetheless be correct.

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