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Relations And Functions

Question
CBSEENMA12032495
Wired Faculty App

Let f : N → Y be function defined as f (x) = 4 x + 3, where, Y = {y ∈N : y = 4 x + 3 for some x ∈ N}. Show that f is invertible. Find the inverse.

Solution

f : N ∴ Y is defined as f(x) = 4 x + 3
where Y = {y ∈ N : y = 4 x + 3 for some x ∈ N }
Consider an arbitrary element y of Y. By the definition of Y, y = 4 x + 3, for some x in the domain N.
therefore space space space space space space space space x equals fraction numerator y minus 3 over denominator 4 end fraction
Define g : Y rightwards arrow N given by g (y) = fraction numerator straight y minus 3 over denominator 4 end fraction
Now,  g o f(x) = g(f(f))= g(4x + 3) = fraction numerator 4 straight x plus 3 minus 3 over denominator 4 end fraction equals x
and   fo g(y) = f(g(y)) = straight f open parentheses fraction numerator straight y minus 3 over denominator 4 end fraction close parentheses equals fraction numerator 4 left parenthesis straight y minus 3 right parenthesis over denominator 4 end fraction plus 3 equals straight y minus 3 plus 3 equals straight y
therefore    g o f = 1N  and f o g = 1y
rightwards double arrow   f is invertible and g is the inverse of f.
therefore           straight f to the power of negative 1 end exponent left parenthesis straight x right parenthesis space equals space fraction numerator straight x minus 3 over denominator 4 end fraction.

 

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}

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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