Cardinality of Set A: 1 or $\le$1?

[ozkan12]

Suppose that set A have unique element...Can we say that cardinality(A)=1 or $cardinality(A)\le1$ ? Which one is true ?

 

[caffeinemachine]

Suppose that set A have unique element...Can we say that cardinality(A)=1 or $cardinality(A)\le1$ ? Which one is true ?

Both are true.

 

[Sudharaka]

Suppose that set A have unique element...Can we say that cardinality(A)=1 or $cardinality(A)\le1$ ? Which one is true ?

Hi ozkan12,

For a finite set Cardinality is the number of elements in the set. If $A$ contains only one element its cardinality is $1$. Saying $\mbox{Cardinality}(A)=1$ is more precise than saying $\mbox{Cardinality}(A)\leq 1$ since the former tells $A$ has exactly one element, whereas the latter tells $A$ has one or no elements.

 

[ozkan12]

First of all, Thank you for your attention...İn your opinion, Which is true ? How both of them is true ?

 

[Sudharaka]

First of all, Thank you for your attention...İn your opinion, Which is true ? How both of them is true ?

You are welcome. This is similar to writing $x=1$. And then saying $x\leq 1$. The $\leq$ sign represents "less than OR equal to". Therefore if $x=1$, it implies that $x\leq 1$ also. In words, $x$ is equal to one and therefore it is also true that $x$ is less than or equal to one.

 

[ozkan12]

That is, we can say that x=1 is more general than $x\le1$

 

[Sudharaka]

That is, we can say that x=1 is more general than $x\le1$

$x=1$ gives more information as it tells us specifically that $x=1$ whereas $x\leq 1$ gives less information; $x$ can be one or it can be less than one.