# Intersection of Rationals and (0 to Infinity)?

• IntroAnalysis
In summary, the intersection of the set of all rational numbers with all positive real numbers is equal to the set of all positive rational numbers. This means that the irrational numbers are not included in this intersection.
IntroAnalysis

## Homework Statement

Let A = [Q$\bigcap$(0,$\infty$)] $\bigcup$ {-1} $\bigcup$(-3, -2]

## Homework Equations

So A = (0,$\infty$) $\bigcup${-1} $\bigcup$(-3,-2]

## The Attempt at a Solution

I understand that the Rational numbers are cardinally equivalent to (0,$\infty$),

but why isn't the intersection of Rationals and (0,$\infty$) =>(0,$\infty$)\Irrationals ?

IntroAnalysis said:
I understand that the Rational numbers are cardinally equivalent to (0,$\infty$),
Quite the contrary, the Irrationals are cardinally equivalent to (0,$\infty$)

The set of all rational numbers is countable, unlike $[0, \infty)$.

Then back to my original question why is the intersection of rationals and (0,∞) = (0,∞)

in other words, why don't irrationals come out of this intersection?

IntroAnalysis said:
Then back to my original question why is the intersection of rationals and (0,∞) = (0,∞)

in other words, why don't irrationals come out of this intersection?

If x belongs to the intersection of A and B, then x belongs to A and x belongs to B. The intersection of rationals and (0,∞) is the set of numbers which are both rationals and positive real numbers.

IntroAnalysis said:
Then back to my original question why is the intersection of rationals and (0,∞) = (0,∞)

in other words, why don't irrationals come out of this intersection?
It isn't. The intersection of the set of all rational numbers with all positive real numbers is all positive rational numbers.

## 1. What is the intersection of rationals and (0 to infinity)?

The intersection of rationals and (0 to infinity) is the set of numbers that are both rational and greater than 0. In other words, it is the set of positive rational numbers.

## 2. How do you represent the intersection of rationals and (0 to infinity) mathematically?

The intersection of rationals and (0 to infinity) can be represented as the set Q ∩ (0, ∞), where Q is the set of rational numbers and (0, ∞) represents all numbers between 0 and infinity, not including 0.

## 3. Is the intersection of rationals and (0 to infinity) an infinite set?

Yes, the intersection of rationals and (0 to infinity) is an infinite set because both the set of rational numbers and the set of numbers between 0 and infinity are infinite sets. Therefore, their intersection must also be infinite.

## 4. Can you give an example of a number that belongs to the intersection of rationals and (0 to infinity)?

One example of a number that belongs to the intersection of rationals and (0 to infinity) is 2.5. This number is rational (as it can be expressed as the ratio of two integers, 5/2) and it is also greater than 0.

## 5. Why is the intersection of rationals and (0 to infinity) important in mathematics?

The intersection of rationals and (0 to infinity) is important in mathematics because it helps us understand the relationship between rational numbers and numbers between 0 and infinity. It also plays a crucial role in many mathematical concepts and applications, such as calculus and number theory.

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