smallest element in Partial ordered set

Question

Every finite partial order has a smallest element, where an element x $\epsilon$ S is said to be the smallest if for all y $\epsilon$ S; it is the case that (x,y)$\epsilon$R. here R is relation defining partial order. How could we can prove or disprove this statement

Answer 1

HINT: Disprove it by finding a counterexample; you can make with as few as two elements. Thinking about Hasse diagrams may help you to visualize what you need. (Of course the empty partial order also provides a counterexample, but I suspect that the finite set was intended to be non-empty.

Answer 2

Hint: For a finite poset with no smallest element, try something like the set $\{1,2,3,4, a, b, c\}$, where you order the numbers in the natural way, and the letters in the natural way, and make no specification about "mixed" couples.

There is no such thing as the smallest element.

You can easily cut this example down to a smaller size if you like.