Which of the following relations in R is an equivalence relatilon-xR-1-y - -x-y-xR-2-y - x - yxR-4-y - x lt-yxR-3-y -x-y

The correct option is A xR_1 y ⇔ |x|=|y|It is simple to check that only, R_1 is an equivalence relation. ...

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Standard IX

Mathematics

Equivalence Relation

Which of the ...

Question

Which of the following relations in R is an equivalence relatilon?

x R_{1} y \Leftrightarrow | x | = | y |

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x R_{2} y \Leftrightarrow x \geq y

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x R_{3} y \Leftrightarrow \frac{x}{y}

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x R_{4} y \Leftrightarrow x < y

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Solution

The correct option is A $x R_{1} y \Leftrightarrow | x | = | y |$
It is simple to check that only, $R_{1}$ is an equivalence relation.

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Similar questions

Q. In the set Z of all integers, which of the following relation R is not an equivalence relation?
(a) x R y : if x ≤ y
(b) x R y : if x = y
(c) x R y : if x − y is an even integer
(d) x R y : if x ≡ y (mod 3)

Determine whether each of the following relations are reflexive, symmetric and transitive:
(i)Relation R in the set A = {1, 2, 3...13, 14} defined as
R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as R = {(x, y): x − y is an integer}

(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
(e) R = {(x, y): x is father of y}

Q. Determine whether each of the following relations are reflexive, symmetric and transitive: (i)Relation R in the set A = {1, 2, 3…13, 14} defined as R = {( x , y ): 3 x − y = 0} (ii) Relation R in the set N of natural numbers defined as R = {( x , y ): y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {( x , y ): y is divisible by x } (iv) Relation R in the set Z of all integers defined as R = {( x , y ): x − y is as integer} (v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {( x , y ): x and y work at the same place} (b) R = {( x , y ): x and y live in the same locality} (c) R = {( x , y ): x is exactly 7 cm taller than y } (d) R = {( x , y ): x is wife of y } (e) R = {( x , y ): x is father of y }

Q. Which one of the following is not a function?

(a) {(x, y) : x, y ∈ R, x² = y}
(b) {(x, y) : x, y ∈, R, y² = x}
(c) {(x, y) : x, y ∈ R, x² = y³}
(d) {(x, y) : x, y ∈, R, y = x³}

Which one of the following is not a function ?

(a) ${(x, y) : x, y ϵ R, x^{2} = y}$
(b) ${(x, y) : x, y ϵ R, y^{2} = x}$
(c) ${(x, y) : x, y ϵ R, x = y^{3}}$
(d) ${(x, y) : x, y ϵ R, y = x^{3}}$

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Which of the following relations in R is an equivalence relatilon?

The correct option is A xR1 y⇔|x|=|y|It is simple to check that only, R1is an equivalence relation.

The correct option is A $x R_{1} y \Leftrightarrow | x | = | y |$
It is simple to check that only, $R_{1}$ is an equivalence relation.