Quote:
Originally Posted by PhantomWolf
The set of "ordinary" real numbers are just as large as the set of complex numbers.
Hmmm, now here's a question. If I have an infinite set and you have an infinite set, but my set also includes all of yours, and also has members that yours doesn't, can it be classed as bigger?
edited to add: Essentially Real numbers are a subset of Imaginary Numbers, and so the set of Imaginary numbers must somehow be bigger, even though they are both infinite. A weird idea, but.......
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Some others have answered your question of my statement, but I thought another example might be helpful.
The set {1,2} is smaller than {1,2,3}, because it is contained in it, right? But what about {1,2} and {2,3}? Neither is contained in the other, but they both have the same number of elements. Similarly, it is obvious that the odd positive integers have the same number of elements as the even positive integers, right? That's because for every odd number, there is an even number (the number just 1 more than it).
But, as
montebianco mentions, we can say the same thing for the integers and the even integers. Although the even integers are completely contained in the integers, there is still one unique even integer for every single integer, and vice versa. That's why we say the sets are the same size: the size of the integers is the same as the size of the even integers.