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Let $\text{N}$ be the set of positive integers. Consider the relation $\text{R}$ on $\text{N}$ defined by $x\text{R}y$ if and only if $\gcd(x,y)>1.$

The relation $\text{R}$ on $\text{N}$ is ________

  1. Reflexive
  2. Symmetric
  3. Transitive
  4. Equivalence relation
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1 Answer

8 votes
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Not transitive because $4\text{R}6$ and $6\text{R}9$ But $4$ is not related to $9.$

$\text{R}$ is not reflexive since $\gcd(1,1)=1$ and $1\text{R}1$ is false.

$\text{R}$ is symmetric since if $x\text{R}y$ then $\gcd(x,y)>1$ and $\gcd(y,x) = \gcd(x,y)>1$ and therefore $y\text{R}x.$

$\text{R}$ is not transitive: $2\text{R}6$ and $6\text{R}3$ are both true but $2\text{R}3$ is false.
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