Can anyone help me explain this to me please? An explanation would be helpful. T

Question

Can anyone help me explain this to me please? An explanation would be helpful. Thank You

were looking at set cardinality and one-to-one correspondence. So, one way to determine how big a set is involves counting up how many items are in the set. For example, if A={a,b,c,d}, then the cardinality of A, |A|=4. This is because there are 4 items listed for A. Another approach is to take the elements in a set and line them up with the natural numbers (or a subset of...). Here's how this could work: A a b c d Natural 1 2 3 4 5 6 7 8 ... #s Notice that we have paired up: 1--a, 2--b, 3--c and 4--d. And then...there's nothing left in A to work with, so were done. This gives us |A|=4 because we can up with a bijective correspondence between a subset of the natural numbers and our set A. O.k. Now...my question is: Can we always do this? What happens if our set A is infinite? How might we establish a bijection between A and the natural numbers...or can't we do it...or does it depend?

Explanation / Answer

It depends on A. actually when you say A is infinite, there are 2 kinds of infinite sets. one is countably infinite and the other is uncountably infinite... For countably infinite set, we can form a bijecton with Natural numbers, where as for uncountable sets, you cannot do that

For example Consider the set A1 = { 1/n : n is a natural number}, this is countably infinite.

Consider the set R = set of real numbers, this is not countably infinte, as we can't form any bijection between R and natural numbers..So R is uncountably infinite..

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