Why ℕ ~ {2x|x∈ℕ}?
I actually understand this but I can’t explain this to my father who says that ‘for each even natural number there are 2 natural numbers’ – How can I explain this concept to him in a way that he will understand? (not saying he is stupid or anything he just cant figure it out)
In: Mathematics
Maybe explain it from this point of view: there’s more than one way to measure the size of sets. When you’re looking at finite sets, all the good ones give you the same answer. But there are different generalizations when you talk about infinity.
In this case, the most common one is talking about cardinality, which is the sense in which the naturals and the even naturals have the same size. Lots of good explanations for this which your dad should understand on this thread.
But, there’s also another way of checking sizes which is a lot more intuitive, I’d say. It’s called natural density. Basically, take the interval [1,n] and check the probability of a random number in your chosen set A of being in that interval. So in our case for evens it’s 1/2. Half the numbers will be even, the other half odd. Now, take n-> infinity. If that probability goes to a certain number (ie, doesn’t fluctuate around 2 or more numbers), then it’s called the natural density of the set A in the naturals. Since it converges to ½, the natural density of the evens is ½, which fits our intuition better.
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