Show that f : N → N, given by f(x)={ x+1, if x is odd, and x-1, if x is even} is both one-one and onto.

Class 12, NCERT Chapter 1, Example12

Suppose f(x₁ ) = f(x₂ ).
Note that if x₁ is odd and x₂ is even, then we will have
x₁ + 1 = x₂ – 1, i.e., x₂ – x₁ = 2, which is impossible.
Similarly, the possibility of x₁ being even and x₂ being odd can also be ruled out, using a similar argument.
Therefore, both x₁ and x₂ must be either odd or even.
Suppose both x₁ and x₂ are odd.
Then f(x₁ ) = f(x₂ )
⇒ x₁ + 1 = x₂ + 1
⇒ x₁ = x₂
Similarly, if both x₁ and x₂ are even, then also
f(x₁ ) = f(x₂ )
⇒ x₁ – 1 = x₂ – 1
⇒ x₁ = x₂ .
Thus, f is one-one.
Also, any odd number 2r + 1 in the co-domain N is the image of 2r + 2 in the domain N and
any even number 2r in the co-domain N is the image of 2r – 1 in the domain N.

Thus, f is onto.

Comments

. Let f : R → R be defined as f(x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto.

Class 12, NCERT Chapter 1, Exercise 1.2, Q12 f : R → R defined as f ( x ) = 3 x . Let x , y ∈ R such that f ( x ) = f ( y ). ⇒ 3 x = 3 y ⇒ x = y ∴ f is one-one. Also, for any real number ( y) in co-domain R , there exists in R such that . ∴ f is onto. Hence, function f is one-one and onto. The correct answer is A.

Let f : N → N be defined by f (n) ={ (n+1)/2, if n is odd and (n-1)/2, if n is even, for all n ∈ N.State whether the function f is bijective. Justify your answer

Class 12, NCERT Chapter 1, Exercise 1.2, Q9 Thus it is bijective.

Show that a one-one function f : {1, 2, 3} → {1, 2, 3} must be onto.

Class 12, NCERT Chapter 1, Example14 Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co-domain {1, 2, 3} under f. Hence, f has to be onto .

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