July 23rd, 2014, 07:29 AM  #1 
Newbie Joined: Jul 2014 From: India Posts: 21 Thanks: 0  Relation
there is a relation defined by xRy iff xy+(2)^(1/2) is an irrational number the R is: 1) an equivalence relation 2) symmetric 3) transitive 4) none of these 
July 23rd, 2014, 08:43 AM  #2 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus 
xRx this means that $xx+\sqrt{2}=\sqrt{2}$ is irrational, so the relation is reflexive. xRy this means that $xy+\sqrt{2}$ is irrational but $yx+\sqrt{2}$ is irrational too so the relation is symmetric. To prove that this relation is not transitive use $x=1$, $y=\sqrt{2}$ and $z=\sqrt{2}$. $1\mathcal{R}\left(\sqrt{2}\right)$, because $1(\sqrt{2})+\sqrt{2}=1+2\sqrt{2}$ is irrational. $\left(\sqrt{2}\right)\mathcal{R}\left(\sqrt{2}\right)$ because $\sqrt{2}\sqrt{2}+\sqrt{2} =\sqrt{2}$ which is irrational but $1\cancel{\mathcal{R}}\sqrt{2}$, $1\sqrt{2}+\sqrt{2}=1$ is not irrational. 

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