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Relations and Functions

Short Answer Type

261.

Find the number of all one-one functions from set A = {1, 2, 3} to itself.

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262. Let f : X → Y be a function. Define a relation R in X given by R = {(a, b) : f (q) = f(b)}. Examine if R is an equivalence relation.

For every a ∈ X. (a. a) ∈ R as f (a) = f (a),
∴ R is reflexive.
Similarly. (a, b) ∈ R ⇒ f(a) = f(b) ⇒ f(b) = f(a) ⇒ (b, a) ∈ R ∴ R is symmetric.
Further, (a, b) ∈ R and (b, c) ∈ R
⇒ f (a) = f (b) and f (b) = f(c) ⇒ f (a) = f (c) ⇒ (a, c) ∈ R ∴ R is transitive.
∴ R is an equivalence relation.

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263.

Let f : Z → Z, g : Z → Z be functions defined by
f = {(n, n²): n ∈ Z} and g = {(n | n |²): n ∈ Z}. Show that f = g.

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264.

Let f : R → R be defined as f (x) = x⁴ 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.

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265.

Let f : R → R be defined as f (x) = 3 x. 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 onc-one nor onto.

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266. Let f : {2, 3, 4, 5} → {3, 4, 5, 9 } and g : {3, 4, 5,9} → {7, 11, 15} be functions defined as f (2) = 3, f (3) = 4, f(4) = f(5) = 5 and g (3) = g (4) = 7 and g (5) = g (9) = 11. Find g of.

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267. Let f : {11, 3, 4} → {1, 2, 5} and g : {1, 2. 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down go f.

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