The relation S defined on the set R of all real number by the rule a S

1.	The relation S defined on the set R of all real number by the rule a Sb iff a ≥ b isA. an equivalence relationB. reflexive, transitive but not symmetricC. symmetric, transitive but not reflexiveD. neither transitive nor reflexive but symmetric
Answer» B. reflexive, transitive but not symmetric S: a S b ⟺ a ≥ b Since a=a ∀a ∈ R, therefore a ≥ a always. Hence (a, a) always belongs to S ∀a ∈ R. Therefore, S is reflexive. If a ≥ b then b ≤ a ⇏ b ≥ a. Hence if (a, b) belongs to S, then (b, a) does not always belongs to S. Hence S is not symmetric. If a ≥ b and b ≥ c, therefore a ≥ c. Hence if (a, b) and (b, c) belongs to S, then (a, c) will belong to S ∀a, b, c∈R. Hence, S is transitive.

The relation S defined on the set R of all real number by the rule a Sb iff a ≥ b isA. an equivalence relationB. reflexive, transitive but not symmetricC. symmetric, transitive but not reflexiveD. neither transitive nor reflexive but symmetric

Answer»

B. reflexive, transitive but not symmetric

S: a S b ⟺ a ≥ b

Since a=a ∀a ∈ R, therefore a ≥ a always. Hence (a, a) always belongs to S ∀a ∈ R. Therefore, S is reflexive.

If a ≥ b then b ≤ a ⇏ b ≥ a. Hence if (a, b) belongs to S, then (b, a) does not always belongs to S. Hence S is not symmetric.

If a ≥ b and b ≥ c, therefore a ≥ c. Hence if (a, b) and (b, c) belongs to S, then (a, c) will belong to S ∀a, b, c∈R. Hence, S is transitive.

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The relation S defined on the set R of all real number by the rule a Sb iff a ≥ b isA. an equivalence relationB. reflexive, transitive but not symmetricC. symmetric, transitive but not reflexiveD. neither transitive nor reflexive but symmetric

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