Show that the relation R in the set A of all the books in a library of a college, given by R = {( x , y ): x and y have same number of pages} is an equivalence relation.

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Solution

The given relation R in the set A of all the books in a library of a college is defined as R={( x,y ):x and y have same number of pages}.

( a,a )∈R, since, the books a and a have the same number of pages. Hence R is reflexive.

Let ( a,b )∈R, thus the books a and b have the same number of pages. Then, ( b,a )∈R since the books b and a also have the same number of pages. Hence, Ris symmetric.

Let, ( a,b ),( b,c )∈R, thus the books a and b, and b and c have the same number of pages. So, the books a and c also have same number of pages. This implies that ( a,c )∈R, hence Ris transitive.

Therefore, the given relation R={( x,y ):x and y have same number of pages} in set A of all the books in a library of a college is reflexive, symmetric and transitive and hence R is an equivalence relation.

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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 }

Show that each of the relation

R

in the set

A = {x \in Z : 0 \leq x \leq 12}

, given by is an equivalence relation. Find the set of all elements related to

1

in each case.

R = {(a, b) : a = b}

Q. Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

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