What if randomly select a number x from the set

Kaeden Woodard

Kaeden Woodard

Answered question

2022-05-22

What if randomly select a number x from the set [0,1], thenP[x is irrational]=1?

Answer & Explanation

Jarrett Reyes

Jarrett Reyes

Beginner2022-05-23Added 8 answers

This can be answered using measure theory. The idea is that, although dense, the measure of rationals in the interval is zero.

Informally, that is:
P ( x [ 0 , 1 ] Q ) = μ ( [ 0 , 1 ] Q ) μ ( [ 0 , 1 ] ) = μ ( [ 0 , 1 ] ) μ ( [ 0 , 1 ] Q ) μ ( [ 0 , 1 ] ) = 1 0 1 = 1.
Even more formally, we use the Lebesgue integral, to integrate the characteristic function
χ Q ( x ) = { 1 , x Q 0 , else
over the interval [ 0 , 1 ]. This is how you are to show that the measure of rationals is 0.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?