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Standard XII

Mathematics

Transitive Relations

Determine whe...

Question

Determine whether each of the following relations are reflexive, symmetric and transitive:

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

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Solution

For reflexive put y=x, x-x =0 which is an integer for all $x ϵ Z$ . So, R is reflexive on Z.
(b)For symmetry let $(x, y) ϵ R$ , then (x-y)is an integer $λ$ and also $y - x = - λ [∵ λ ϵ Z \Rightarrow - λ ϵ Z]$
$∴ y - x$ is an integer $\Rightarrow (y, x) ϵ R$ . SO, R is symmetric.
(c)For transitivity let (x,y) belongs R and $(y, z) ϵ R ∴ x - y$ = integer and y-z = integer, then x-z is also an integer
$∴ (x, z) ϵ R$ . So, R is transitive

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Q. Relation

R

in the set

Z

of all integers defined as

R = {(x, y) : (x - y) i s a n i n t e g e r}

enter 1-reflexive and transitive but not symmetric

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) Relative

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}

R = {(x, y) : y is divisible by x}

(iv) Relative

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}

Determine whether each of the following relations are reflexive, symmetric and transitive:
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}

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Determine whether each of the following relations are reflexive, symmetric and transitive: (iv) Relation R in the set Z of all integers defined as R = {(x, y): x − y is an integer}

Determine whether each of the following relations are reflexive, symmetric and transitive:

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