Let R be the relation on the set of all real numbers defined by a R b iff - a - b - - 1- Then R isA- Symmetric onlyB- Transitive onlyC- Reflexive and SymmetricD- Anti-symmetric only

The correct option is C Reflexive and Symmetric|a a| = 0 lt; 1 ∴ a R a ∀ a ϵ R ∴ R is reflexive. Again a R b ⇒ |a b| ≤ 1 ⇒ |b a| ≤ 1 ⇒ b R a ∴ R is sy ...

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Standard X

Mathematics

Reflexive Relations

Let R be the ...

Question

Let R be the relation on the set of all real numbers defined by a R b iff $| a - b | \leq 1$ . Then R is

Reflexive and Symmetric

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Symmetric only

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Transitive only

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Anti-symmetric only

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Solution

The correct option is A Reflexive and Symmetric
$| a - a | = 0 < 1 ∴ a R a \forall a ϵ R$
$∴$ R is reflexive.
Again $a R b \Rightarrow | a - b | \leq 1 \Rightarrow | b - a | \leq 1 \Rightarrow b R a$
$∴ R$ is symmetric.
Further, 1 R 2 and 2 R 3 but $1 / R 3, [∵ | 1 - 3 | = 2 > 1]$
$∴ R$ is not transitive.

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Let R be the relation on the set of all real numbers defined by a R b iff |a−b|≤1. Then R is

The correct option is A Reflexive and Symmetric|a−a|=0<1∴aRa∀aϵR ∴ R is reflexive. Again aRb⇒|a−b|≤1⇒|b−a|≤1⇒bRa ∴R is symmetric. Further, 1 R 2 and 2 R 3 but 1/R3,[∵|1−3|=2>1] ∴R is not transitive.

Let R be the relation on the set of all real numbers defined by a R b iff $| a - b | \leq 1$ . Then R is