Finite Sets

[kim124578]

My math book has a definition and I'm a little confused. It says a set A is finite if and only if there is a one-to-one function f on A into N_k (a subset of the naturals read as N sub k). If I have a set A = {1, 2, 3, 3, 3, 4}, it's clearly finite with 6 elements, but it has the number 3 repeated three times. A 1-1 function f is a function such that f(i) = f(j) only when i=j. I guess I'm missing something incredibly simple, my questions are

1. Does there exist a 1-1 function of A, if so, can you give an example of this function?
2. Is A finite?

 

[pka]

My math book has a definition and I'm a little confused. It says a set A is finite if and only if there is a one-to-one function f on A into N_k (a subset of the naturals read as N sub k). If I have a set A = {1, 2, 3, 3, 3, 4}, it's clearly finite with 6 elements, but it has the number 3 repeated three times. A 1-1 function f is a function such that f(i) = f(j) only when i=j. I guess I'm missing something incredibly simple, my questions are
1. Does there exist a 1-1 function of A, if so, can you give an example of this function?
2. Is A finite?

Was the example \(A = \{1, 2, 3, 3, 3, 4\}\) given by your textbook as a finite set?
If so, I would object because it there are repeated terms. Usually one see the definition that a set \(S\) is finite if and only there is no one-to-one function from \(S\) to \(S\) that is not onto If we take \(\mathbb{N}=\{0,1,2,3,\cdots\}\) then the function \(f(n)=2\cdot n\) one-to-one function from \(\mathbb{N}\to\mathbb{N}\) that is not onto. Thus by definition \(\mathbb{N}\) is not finite.

 

[Dr.Peterson]

My math book has a definition and I'm a little confused. It says a set A is finite if and only if there is a one-to-one function f on A into N_k (a subset of the naturals read as N sub k). If I have a set A = {1, 2, 3, 3, 3, 4}, it's clearly finite with 6 elements, but it has the number 3 repeated three times. A 1-1 function f is a function such that f(i) = f(j) only when i=j. I guess I'm missing something incredibly simple, my questions are

1. Does there exist a 1-1 function of A, if so, can you give an example of this function?
2. Is A finite?

First, as pka pointed out, your set notation is bad; [MATH]A[/MATH] should be written as [MATH]\{1,2,3,4\}[/MATH], since no element can be repeated. It has only 4 elements. (I don't think you are studying multisets.)

Second, I'm guessing that your [MATH]N_k[/MATH] is not just any subset of N, but specifically the set [MATH]\{k\in \mathbb{N} | k\le n\}[/MATH]. If it could be any subset, then the definition would be wrong. If it is as I suggest, it is a valid (and elementary) definition.

To answer your questions then, [MATH]A[/MATH] is in fact identical to [MATH]N_4[/MATH], and is finite. (I'm assuming, by the way, that you define the natural numbers to exclude 0.)

 

[Steven G]

To be clear set A = {1, 2, 3, 3, 3, 4} = {1, 2, 3, 4} and for cardinality we use use the set that does not have duplicates in it. | A | = 4