Let N be the set of integers. A relation R on N is defined as R = {(x,

1.	Let N be the set of integers. A relation R on N is defined as R = {(x, y) : xy > 0, x, y ∈ N}. Then, which of the following is correct ?(a) R is symmetric but not reflexive (b) R is reflexive but not symmetric (c) R is symmetric and reflexive but not transitive (d) R is an equivalence relation
Answer» (d) R is an equivalence relation R = {(x, y) : xy > 0 x, y ∈N} • x, x ∈N ⇒ x2 > 0 ⇒ R is reflexive • x, y ∈N and (x, y) ∈R ⇒ xy > 0 ⇒ yx > 0 (y, x) ∈R ⇒ R is symmetric • x, y, z ∈N and (x, y) ∈R, (y, z) ∈R ⇒ xy > 0 and yz > 0 ⇒ xz > 0 ⇒ (x, z) ∈R ⇒ R is transitive. ∴ R is an equivalence relation.

Let N be the set of integers. A relation R on N is defined as R = {(x, y) : xy > 0, x, y ∈ N}. Then, which of the following is correct ?(a) R is symmetric but not reflexive (b) R is reflexive but not symmetric (c) R is symmetric and reflexive but not transitive (d) R is an equivalence relation

Answer»

(d) R is an equivalence relation

R = {(x, y) : xy > 0 x, y ∈N}

• x, x ∈N ⇒ x2 > 0 ⇒ R is reflexive

• x, y ∈N and (x, y) ∈R ⇒ xy > 0

⇒ yx > 0 (y, x) ∈R

⇒ R is symmetric

• x, y, z ∈N and (x, y) ∈R, (y, z) ∈R

⇒ xy > 0 and yz > 0

⇒ xz > 0 ⇒ (x, z) ∈R

⇒ R is transitive.

∴ R is an equivalence relation.

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Let N be the set of integers. A relation R on N is defined as R = {(x, y) : xy &gt; 0, x, y ∈ N}. Then, which of the following is correct ?(a) R is symmetric but not reflexive (b) R is reflexive but not symmetric (c) R is symmetric and reflexive but not transitive (d) R is an equivalence relation

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Let N be the set of integers. A relation R on N is defined as R = {(x, y) : xy > 0, x, y ∈ N}. Then, which of the following is correct ?(a) R is symmetric but not reflexive (b) R is reflexive but not symmetric (c) R is symmetric and reflexive but not transitive (d) R is an equivalence relation