Question

To find the pair of elements \((7,7)\) are comparable in the poset \(\left( {{z^ + },1} \right)\).

Short answer

The pair of elements \((7,7)\) are comparable in the poset \(\left( {{z^ + },1} \right)\).

Step-by-step solution

Given data

The pair of elements \((7,7)\).

Concept used of partially ordered set

A relation\(R\)is a poset if and only if,\((x,x)\)is in\({\rm{R}}\)for all x (reflexivity)

\((x,y)\)and\((y,x)\)in R implies\(x = y\)(anti-symmetry),\((x,y)\)and\((y,z)\)in R implies\((x,z)\)is in\({\rm{R}}\)(transitivity).

Compare the pair of elements

Now consider the pair of elements \((7,7)\).

As \(7\mid 7 \Rightarrow \) The pair of elements \((7,7)\) are comparable in the poset \(\left( {{z^ + },1} \right)\).

Therefore, the pair of elements \((7,7)\) are comparable in the poset \(\left( {{z^ + },1} \right)\).