Question

There is a set $A$ possessing the following property
$"\forall x\in A\exists y\in A$ such that $y>x."$ This means

• A. $y$ is always greater than $x.$
  
• B. $x$ is the least element of $A.$
  
• C. $A$ has no maxima.
  
• D. Each element of the set is bigger than some other element.

Answer 1

The correct option is C $A$ has no maxima.


Clearly, set $A$ has no greatest element as there is an element $y\forall x\in A$ such that $y>x$.
Therefore, $y$ is not always greater than $x$, hence $A$ has no maxima.
Similarly, we cannot find a least element, hence, $x$ cannot be certainly the least element of $A$.
We can say each element is smaller than any other element but not vice-versas, hence we cannot say each element is bigger than some other element.