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Relations And Functions

Question
CBSEENMA12032518
Wired Faculty App

In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 1 + x2

Solution

f : R → R defined by
f(x) = 1 + x2
Let  straight x subscript 1 comma space straight x subscript 2 space element of space straight R space such that straight f left parenthesis straight x subscript 1 right parenthesis equals straight f left parenthesis straight x subscript 2 right parenthesis


rightwards double arrow space space space 1 plus straight x subscript 1 superscript 2 space equals space 1 plus straight x subscript 2 superscript 2 rightwards double arrow space space space straight x subscript 1 superscript 2 space equals space straight x subscript 2 superscript 2 rightwards double arrow space space space straight x subscript 1 equals plus-or-minus space straight x subscript 2
therefore space space space straight f left parenthesis straight x subscript 1 right parenthesis space equals space straight f left parenthesis straight x subscript 2 right parenthesis does not imply that x1 = x2 for instance
f(1) = (f - 1) = 2
therefore  f is not one-one
Consider an element of  - 2 in co domain R.
It is seen that f(x) = 1 + x2 is positive for all x element of straight R
Thus there does not exists any x in domain R such that  f(x) = - 2
therefore    f is not onto
Hence, f is neither one-one nor onto.

Some More Questions From Relations and Functions Chapter

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.

If R and R’ arc reflexive relations on a set then so are R ∪ R’ and R ∩ R’. 

If R and R’ are symmetric relations on a set A, then R ∩ R’ is also a sysmetric relation on A.

Show that the union of two symmetric relations on a set is again a symmetric relation on that set.

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