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Question

Let $R subscript 1 end subscript$ be a relation defined by
$R subscript 1 end subscript equals open curly brackets open parentheses a comma b close parentheses vertical line a greater or equal than b comma a comma b element of R close curly brackets.$ Then, $R subscript 1 end subscript$ is

An equivalence relation on $R$
Reflexive, transitive but not symmetric
Symmetric, transitive but not reflexive
Neither transitive not reflexive but symmetric

The correct answer is: Reflexive, transitive but not symmetric

For any $a element of R comma$ we have $a greater or equal than a$
Therefore, the relation $R$ is reflexive.
$R$ is not symmetric as $left parenthesis 21 right parenthesis element of R$ but $open parentheses 12 close parentheses not an element of R.$ The relation $R$ is transitive also, because $open parentheses a comma b close parentheses element of R comma left parenthesis b comma c right parenthesis element of R$ imply that $a greater or equal than b$ and $b greater or equal than c$ which in turn imply that $a greater or equal than c$

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