Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d ) ⇔ a d (b + c) = b c (a + d). Check whether R is an equivalence relation on N x N.

Solution

(i) Let (a, b) be any element of N x N
Now (a, b) ∈ N x N ⇒ a, b ∈ N ∴ a b (b + a) = b a (a + b)
⇒ (a, b) R (a, b)
But (a, b) is any element of N x N ∴ (a, b) R (a, b) ∀ (a, b) ∈ N x N ∴ R is reflexive on N x N.
(ii) Let (a, b), (c, d ) ∈ N x N such that (a, b) R (c, d)
Now (a, b) R (c, d) ⇒ a d (b + c) = b c (a + d)
⇒ c b (d + a) = d a (c + b)
⇒ (c, d) R (a, b)
∴ (a, b) R (c, d ) ⇒ (c, d) R (a, b) ∀ (a, b), (c, d) ∈ N x N ∴ R is symmetric on N x N.
(iii) Let (a, b), (c, d ), (e, f) ∈ N x N such that
(a, b) R (c, d ) and (c, d ) R (e, f)
(a, b) R (c, d) ⇒ a d (b + c) = b c (a + d)
$rightwards double arrow space space fraction numerator straight b plus straight c over denominator bc end fraction equals fraction numerator a plus d over denominator a d end fraction space rightwards double arrow space 1 over b plus 1 over c equals 1 over a plus 1 over d space space space space space space space space space space space space space... left parenthesis 1 right parenthesis$
Also (c, d) R (e, f) $rightwards double arrow$ cf(d + c) = d e (c + f)
$rightwards double arrow space space fraction numerator straight d plus straight c over denominator dc end fraction equals fraction numerator straight c plus straight f over denominator cf end fraction rightwards double arrow 1 over straight d plus 1 over straight c equals 1 over straight c plus 1 over straight f space space space space space space space space space space space space space space space space space space space space space space space space space.. left parenthesis 2 right parenthesis$
Adding (1) and (22), we get
$open parentheses 1 over straight b plus 1 over straight c close parentheses plus open parentheses 1 over straight d plus 1 over straight c close parentheses equals open parentheses 1 over straight a plus 1 over straight d close parentheses plus open parentheses 1 over straight c plus 1 over straight f close parentheses rightwards double arrow space space space 1 over straight b plus 1 over straight c equals 1 over straight a plus 1 over straight f space space space space rightwards double arrow space space space space fraction numerator straight b plus straight c over denominator ce end fraction equals fraction numerator straight a plus straight f over denominator af end fraction$

⇒ a f (b + e) = b e (a + f) ⇒ (a, b) R (e,f)

∴ (a, b) R (c, d) and (c, d) R (e.f) ⇒ (a, b) R (e, f) ∀ (a, b), (b, c), (c, d) ∈ N x N ∴ R is transitive on N x N ∴ R is reflexive, symmetric and transitive ∴ R is an equivalence relation on N x N

Some More Questions From Relations and Functions Chapter

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

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 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}

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Relations And Functions

Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d ) ⇔ a d (b + c) = b c (a + d). Check whether R is an equivalence relation on N x N.

Some More Questions From Relations and Functions Chapter

If a matrix has 24 elements, what are the possible orders it can have ? Wh'at. if it has 13 elements ?

lf a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.

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For Daily Free Study Material Join wiredfaculty

Relations And Functions

Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d ) ⇔ a d (b + c) = b c (a + d). Check whether R is an equivalence relation on N x N.

Some More Questions From Relations and Functions Chapter

If a matrix has 24 elements, what are the possible orders it can have ? Wh'at. if it has 13 elements ?

lf a matrix has 18 elements, what are the possible orders it can have ? What, if it has 5 elements?

If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a, b) : a is sister of b} is the empty relation and R’ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive but neither symmetric nor transitive.

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Give an example of a relation which is (i) Symmetric but neither reflexive nor transitive.(ii) Transitive but neither reflexive nor symmetric.(iii) Reflexive and symmetric but not transitive.(iv) Reflexive and transitive but not symmetric.(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L1, L2) : L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Give an example of a relation which is

(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Let L be the set of all lines in a plane and R be the relation in L defined as R = {(L₁, L₂) : L₁ is perpendicular to L₂}. Show that R is symmetric but neither reflexive nor transitive.