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
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.
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