Relations And Functions

Question

Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).

Then

(i) Find the identify element of (A, )
(ii) Find the invertible elements of (A, )

Solution

(i) Let (x, y) be the identify element of (A, *).
∴ (a, b) * (x,y) = (a, b) ∀ a, b ∈ Q ⇒ (ax, ay + b) = (a, b)
⇒ ax = a, ay + b = b ⇒ ax = a, ay = 0 ∀ a ∈ Q ⇒ x = 1, y = 0
Now (a, b) * (1,0) = (a, b) ∀ a, b, ∈ Q Also (1,0)* (a, b) = (a, 1. b + 0) = (a, b)
∴ (1, 0) is the identity element of A.
(ii) Let (a, b) ∈ A be invertible ∴ there exists (c, d) ∈ A such that (a, b) * (c, d) = (1, 0) = (c, d) * (a, b)
∴ (ac, ad + b) = ( 1,0) ⇒ ac = 1, ad + b = 0
$rightwards double arrow space space space straight c equals 1 over straight a comma space space space straight d equals fraction numerator negative straight b over denominator straight a end fraction$

Now a ≠ 0
∵ if a = 0, then (0, b) is not invertible as (0, b) invertible implies that here exists (c, d)∈ A such that (0, b) * (c, d) = (1, 0) or (0, b) = (1, 0), which is senseless.
$therefore space space space open parentheses 1 over straight a comma space fraction numerator negative straight b over denominator straight a end fraction close parentheses asterisk times left parenthesis straight a comma straight b right parenthesis equals left parenthesis 1 comma 0 right parenthesis rightwards double arrow space space space left parenthesis straight a comma straight b right parenthesis to the power of negative 1 end exponent equals open parentheses 1 over straight a comma space fraction numerator negative straight b over denominator straight a end fraction close parentheses$
$therefore$ invertible elements of A arc (a, b), a $not equal to 0$ and (a, b)^-1 = $open parentheses 1 over straight a comma fraction numerator negative straight b over denominator straight a end fraction close parentheses.$

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 A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).

Then

(i) Find the identify element of (A, )
(ii) Find the invertible elements of (A, )

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.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.

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For Daily Free Study Material Join wiredfaculty

Relations And Functions

Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b). Then (i) Find the identify element of (A, *)(ii) Find the invertible elements of (A, *)

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.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b3} is refleive, symmetric or transitive.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).

Then

(i) Find the identify element of (A, )
(ii) Find the invertible elements of (A, )

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.

Check whether the relation R in R defined by R = {(a,b) : a ≤ b³} is refleive, symmetric or transitive.