# question on finite sets

• May 20th 2009, 06:32 AM
klmsurf
question on finite sets
I'm reading through a book on Real Analysis (Intro.) and got a question on finite sets. The definition I'm using for a finite set is: a set A is finite iff there is a one-to-one function f on Nk onto A (Nk is read N sub k where k is a subscript and Nk is a subset of the natural numbers from 1 to k) for some k.

Question
Let A = {1, 2, 3, 2}, Nk = {1, 2, 3, 4}, and assume that f is bijective from Nk onto A. Observe that f(2) = f(4) implies 2 ≠ 4. It appears that f is not bijective. Is A finite or not? If A is finite, could you describe a bijective function f from Nk onto A.
Thanks
• May 20th 2009, 06:36 AM
TheAbstractionist
Quote:

Originally Posted by klmsurf
I'm reading through a book on Real Analysis (Intro.) and got a question on finite sets. The definition I'm using for a finite set is: a set A is finite iff there is a one-to-one function f on Nk onto A (Nk is read N sub k where k is a subscript and Nk is a subset of the natural numbers from 1 to k) for some k.

Question
Let A = {1, 2, 3, 2}, Nk = {1, 2, 3, 4}, and assume that f is bijective from Nk onto A. Observe that f(2) = f(4) implies 2 ≠ 4. It appears that f is not bijective. Is A finite or not? If A is finite, could you describe a bijective function f from Nk onto A.
Thanks

Hi klmsurf.

Yes, $\displaystyle A$ is finite, but you want to take $\displaystyle N_3=\{1,2,3\},$ not $\displaystyle N_4.$ Then f is a bijection $\displaystyle A\to N_3$ with, e.g., $\displaystyle 1\mapsto1,\ 2\mapsto2,\ 3\mapsto3.$
• May 20th 2009, 08:46 AM
klmsurf
If a bijection exists between two sets, we say they have the same cardinality. A = {1, 2, 3, 2} has exactly four elements, regardless of the distinctness between the elements. Can you explain why the cardinality of A = {1, 2, 3, 2} is 3? Maybe I'm missing a definition or proposition on sets.
• May 20th 2009, 09:03 AM
Plato
Quote:

Originally Posted by klmsurf
If a bijection exists between two sets, we say they have the same cardinality. A = {1, 2, 3, 2} has exactly four elements, regardless of the distinctness between the elements. Can you explain why the cardinality of A = {1, 2, 3, 2} is 3?

$\displaystyle A = \{1, 2, 3, 2\}= \{1, 2, 3\}$ it just has an element listed twice.
$\displaystyle B = \{c,c,c,d,c,d\}= \{c,d\}$ has only two elements.

May I suggest that you study a foundations of mathematics text before you try Intro to Analysis.