**Explanation:**

The set {2x | x is any positive rational number} is an infinite set. Here's why:

**Definition of an Infinite Set:**
An infinite set is a set that has an uncountable number of elements. It cannot be exhausted or enumerated completely.

**Definition of Positive Rational Numbers:**
Positive rational numbers are numbers that can be expressed as a fraction of two positive integers. For example, 1/2, 3/4, and 5/6 are all positive rational numbers.

**Definition of Set Builder Notation:**
The set {2x | x is any positive rational number} is defined using set builder notation. This notation states that for each positive rational number, x, we can obtain an element in the set by multiplying x by 2.

**Proof that the Set is Infinite:**
To show that the set {2x | x is any positive rational number} is infinite, we need to demonstrate that there is no limit to the number of elements in the set.

Let's assume the opposite, that the set is finite. This means that there is a finite number of positive rational numbers, which we know is not true.

Since positive rational numbers can be expressed as fractions, we can always find a new positive rational number by adding 1 to the denominator of a fraction. For example, if we have 1/2, we can get 1/3, 1/4, 1/5, and so on, by adding 1 to the denominator.

Since there is no limit to the number of positive rational numbers, there is also no limit to the number of elements in the set {2x | x is any positive rational number}.

Therefore, the set {2x | x is any positive rational number} is an infinite set.

There are infinitely many positive real numbers. If we tend to substitute all of them for x,we get infinite number of solutions for 2x and hence the set will has infinite number of elements