In the set X = {2, 3, 4, 5, 9, 16, 25, 27, 64, 81, 125} was introduced journal R is defined as follows: aRb exists a natural number k such that b = a^k. Draw a graph of the relationship. Investigate whether it is a partial order relationship (ie, whether it is reflexive, antisymmetric and transitive).
Can You help mi with this ? thanks so much.
Go back to definitions,what does it mean to be reflexive,antisymetric and transitive?
For reflexivity consider any element $a$ of set $X$,then there is natural number k(namely number 1) such that $a=a^k$ thus it is reflexive
For antisymmetry consider and $aRb$ and $bRa$,they imply that $b=a^{k_0}$ and that $a=b^{k_1}$ then $b=(b^{k_1})^{k_0} = b^{k_0k_1}$,since both $k_0,k_1$ are natural they must be both equal to one thus we conclude that $a=b$
For transitivity consider any pair of elements $aRb$ and $bRc$ then $b=a^{k_0}$ and $c=b^{k_1}$,then $c=(a^{k_0})^{k_1}=a^{k_0k_1} $ since $k_0k_1 \in \mathbb{N}$ then we conclude that $aRc$
And voila it is proven that R is partial ordering