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$A=\{1,2,3,4\}$. Determine with reasons whether $R$ is reflexive, symmetric or transitive.

$R=\left\{(1,1),(1,2),(2,1),(2,2)\right\}$

How is this done?

Reflexive must contain every element to itself. Therefore it is not reflexive as there is no $(3,3)$ and $(4,4)$? Or is it the same as symmetric because $(1,2)$ and $(2,1)$ are elements of $R$.

Not transitive because there are not three elements with a transitive link e.g. $(1,2)$, $(2,3)$ and $(1,3)$.

Ben Millwood
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Michael
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1 Answers1

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You are right for symmetry and reflexivity, but wrong for transitivity. Indeed transitivity means: for every $(a,b)$ and $(b,c)$ in $A\times A$, $\textbf{IF}$ $(a,b)$ and $(b,c)$ are in $R$, then $(a,c)$ is in R. So not-transitive means that there exist $(a,b)$ and $(b,c)$ in $R$ such that $(a,c)$ is not in $R$. Now, can you find such $(a,b)$ and $(b,c)$ in $R$, with $(a,c)$ not in $R$? No, you can't, so $R$ is transitive.

Federica Maggioni
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