Why are the natural numbers and the even natural numbers equinumerous?

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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

5 Answers

Anonymous 0 Comments

A good metric to look at is if a set has at least as many elements as another set. This can be checked, so that A has at least as many elements as B if you can pair their elements up and still have some leftover in A but not in B. Your dad will probably agree to this.

So, let’s say A=ℕ and B={2x|x∈ℕ} (so I can write it down more easily). If we pair every n in B up with the same number in A, then we have leftover in A but not in B, so A has at least as many elements as B. If we pair every n in A up with 4*n in B, then we’ll have leftover in B but not in A, so B has at least as many elements as A.

As both sets have at least as many elements as the other, it follows that they’re equal. If your dad still disagrees, ask him which step of this progress is supposedly incorrect.

Anonymous 0 Comments

First of all, argue that if we have two sets and are able to map each element of one set to exactly one element of the other set and vice versa, then the sets are equinumerous. This is quite obvious and easy to prove; if you have X dots on one half of a piece of paper and Y dots on the other, and are able to draw lines connecting each dot with exactly one other on the other side of the paper, then obviously X = Y.

Afterwards, assume the set of natural numbers and the set of even natural numbers. Connect each element from the first set to its double on the second set. Since you have successfully established a bijective mapping between the two sets, they are indeed equinumerous.

If your father is still not convinced, remind him that we are talking about infinitely many numbers; we will always be able to map each number of a set with exactly one number of the other set, simply because the numbers never end. If we had the finite sets {1,2,3,4} and {2,4}, then this obviously wouldn’t be the case.

Anonymous 0 Comments

>”For every even natural number there are 2 natural numbers”

Really? Because I can pair every natural number with an even natural number, and there will be no number left over in either set.

0 => 0

1 => 2

2 => 4

3 => 6

4 => 8

5 => 10

and so on.

We say two sets have the same cardinality if, for each element in one set, we can find a matching element in the other set with no numbers left over.

Ask your father what natural number is left over once all the even natural numbers have been assigned a partner.

Or, if you want to really do his head in, explain that while natural numbers and even natural numbers are the same cardinality, real numbers and natural numbers don’t. There are infinity more real numbers than natural numbers.

Anonymous 0 Comments

Perhaps an important piece of context for the correct and good proofs being offered here is that they’re all oriented around the ways we *choose* to make infinity mathematically comprehensible. Math is a constructed system of definitions. This is likely a case where something is true according to a mathematical definition, but the definition itself feels “wrong.”

The very intuitive approach to comparing the “size” of two infinite sets is to truncate them both at the “same place” and compare the size of the two resulting finite sets. I’d wager that’s what your father is doing in his head. The problem with this is that we’ve jettisoned any notion that the sets are infinite and so aren’t really solving the problem we set out to solve.

Instead mathematicians have come up with a different way of comparing the size of infinite sets that involves asking the question of whether you can map the elements of one onto the elements of the other. Using this definition of “size,” the natural numbers and the even natural numbers have the same size. This is quite counterintuitive, and one might even say that’s a problem with the definition. However, the definition is useful in other contexts, like showing how and why the real numbers are “bigger” than the natural numbers, so we continue to use it.

Anonymous 0 Comments

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.