Cardinality

This page discusses cardinality, a basic topic found in classes like Sets, Relations and Functions and also Real Analysis. Definitions with discussion follow.

numerically equivalent

two sets are numerically equivalent if there exists a one-to-one function from one set onto the other. We can also say two sets have the same cardinal number if this is true. </dl>

If we can say that a set has $n$ elements where $n$ is a specific element of the natural numbers, then we can say that particular set is countable. If each element in a infinite set can be paired with a natural number we say the set is countably infinite and has a cardinal number $\aleph_0$. The natural numbers have the cardinal number $\aleph_0$ . If we pair each element in a set $A$ with a natural number in a one-to-one manner as listed above we call this an enumeration of $A$.

We write card $A$ $\leq$ card $B$ if there is a one-to-one function from $A$ into $B$. The cardinality of the natural numbers is equal to the cardinality of the integers which may not seem logical since there seem to be twice as many integers as natural numbers. However, we can create a 1-1 function which shows us this is true. If $n$, a natural number, is odd, then the function adds 1 to it and divides the result by 2. If $n$ is even, then the function divides $n$ by two and makes it negative. This is a function that maps $\mathbb{N} \rightarrow \mathbb{Z}$. This function is onto so we can see that this function satisfies the definition of numerically equivalent. Therefore, the two sets to have the same cardinal number so we write card $\mathbb{Z} = \aleph_0$.