Question

Let us define a relation $R$ in $R$ as $aRb\text{if}a\ge b.$ Then R is

• A. neither transitive nor reflexive but symmetric
• B. an equivalence relation
• C. reflexive, transitive but not symmetric
• D. symmetric transitive but not reflexive

Answer 1

The correct option is C reflexive, transitive but not symmetric

Given: $aRb\text{if}a\ge b$

Check reflexivity
For reflexive relation $aRa,\forall a\in R$

Here we have $aRb\text{if}a\ge b$

Let $aRa\Rightarrow a\ge a$ which is true.
So, R is reflexive

Check symmetricity

For symmetric relation

If $aRb$ then $bRa,\forall a,b\in R$

Here $(2,1)\in R\text{as}2\ge 1,\text{but}(1,2)\notin R,as1\ngeqq 2$

So R is not symmetric

Check transitivity

​​​​For transitive relation

If $aRb\text{and}bRc\Rightarrow aRc\forall a,b,c\in R$

Let $aRb\text{and}bRc\Rightarrow a\ge b\text{and}b\ge c$

Hence, we can conclude that $a\ge c\Rightarrow aRc$

So, R is transitive
Hence, option (B) is correct one.