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Let A = {1, 2, 3, 4}. Let f: A → A and g: A → A, defined by f = {(1, 4), (2, 1), (3, 3), (4, 2)} and g = {(1, 3), (2, 1), (3, 2), (4, 4)}.

Find (i) g o f (ii) f o g (iii) f o f

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We know that

range f = {1, 2, 3, 4} and domain (g) = {1, 2, 3, 4}

So range (f) ⊆ domain (g)

(i) We know that domain (g o f) = domain (f) = {1, 2, 3, 4}

By substituting the values

(g o f) (1) = g{f(1)} = g(4) = 4

(g o f) (2) = g{f(2)} = g(1) = 3

(g o f) (3) = g{f(3)} = g(3) = 2

(g o f) (4) = g{f(4)} = g(2) = 1

Therefore, g o f = {(1, 4), (2, 3), (3, 2), (4, 1)}

(ii) We know that domain (f o g) = domain (g) = {1, 2, 3, 4}

By substituting the values

(f o g) (1) = f{g(1)} = f(3) = 3

(f o g) (2) = f{g(2)} = f(1) = 4

(f o g) (3) = f{g(3)} = f(3) = 2

(f o g) (4) = f{g(4)} = f(4) = 2

Therefore, f o g = {(1, 3), (2, 4), (3, 2), (4, 2)}

(iii) We know that domain (f o f) = domain (f) = {1, 2, 3, 4}

By substituting the values

(f o f) (1) = f{f(1)} = f(4) = 2

(f o f) (2) = f{f(2)} = f(1) = 4

(f o f) (3) = f{f(3)} = f(3) = 4

(f o f) (4) = f{f(4)} = f(2) = 1

Therefore, f o f = {(1, 2), (2, 4), (3, 3), (4, 1)}

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