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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.

f : N – {1} → N is defined by
f (n) = the highest prime factors of n.
∴ f (6) = the highest prime factor of 6 = 3 f (12) = the highest prime factor of 12 = 3 Now 6 and 12 are associated to the same element.
∴ f is not one-to-one Also range of f consists of prime numbers only ∴ range of f ≠ N ∴ f is not onto function.
Range of f is the set-of all prime numbers.

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