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Let A = {a 1 ,a 2 .....,a n } where n is finite. Range of f = {(f (a 1 ) , f (a 2 ), ...., f (a n )} Since f : A → A is an onto function ∴ range of f = A ⇒ {f (a 1 ),f(a 2 ),...f(a n )} = (a 1 ,a 2 ,....,a n } But A is a finite set consisting of n elements ∴ f(a n ), f(a 2 ), ... ,f(a n ) are district elements of A. ∴ f is one-to-one.

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Relations and Functions

Short Answer Type

251. Prove that the greatest integer function f : R → R, given by f (x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

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252. Let A and B be two sets. Show that f :A x B → B x A such that f (a,b) = (b,a) is a bijective function.

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253.

Consider the identity function 1_N : N → N defined as l_N(x) = x ∀ x ∈ N. Show that although I_N is onto but I_N + I_N : N → N defined as (I_N + I_N) (x) = I_N(x) + I_N(x) = x + x = 2 x is not onto.

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254. Show that the function f : N → N given by f(1) = f (2) = 1 and f (x) = x – 1, for every x > 2, is onto but not one-one.

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255. Show that an onto function f : {1,2,3} → {1, 2, 3} is always one-one.

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

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257.

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

(i) f : R → R defined by f (x) = 3 – 4 x
(ii) f : R → R defined by f (x) = 1 + x².

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258.

Let f : N – {1} → N defined by f (n) = the highest prime factor of n. Show that f is neither one-to-one nor onto. Find the range of f.

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259. Let A be a finite set. If f : A → A is onto, show that f is one-to-one.

Let A = {a₁,a₂.....,a_n} where n is finite.
Range of f = {(f (a₁) , f (a₂), ...., f (a_n)}
Since f : A → A is an onto function
∴ range of f = A
⇒ {f (a₁),f(a₂),...f(a_n)} = (a₁,a₂,....,a_n}
But A is a finite set consisting of n elements
∴ f(a_n), f(a₂), ... ,f(a_n) are district elements of A.
∴ f is one-to-one.

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260. Let A be finite set. If : A → A is one-to-one, then show that f is onto.

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