Doubt

Question

State True/False(with reason).

1. Every non-empty set of non-negative integers has a least element.

2. Every non-empty set of non-negative rationals has a least element.

3. Every non-empty set of integers has a least element.

Answer 1

1. TRUE

    {1,2,3,4,5.......$\infty$} here 1 is the least element.

(we select smallest element of the chain to be the least element in a poset)

2. FALSE

Since there is no smallest positive rational number in a set of non negative rationals so there would be no least element (but if the set is finite then we can compare them and could find the least element)

3. FALSE

   { $\infty$ ....,-3,-2,-1,0,1,2,3.... $\infty$ }

  Since the list is infinite from both sides so there is not any least element