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Question 11

 Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f –1 and show that (f –1)–1 = f

Answer

Function f: {1, 2, 3}  →  {abc} is given by,

f(1) = af(2) = b, and f(3) = c

If we define g: {abc}  →  {1, 2, 3} as g(a) = 1, g(b) = 2, g(c) = 3, then we have:

gof = Ix and fog = Iy where X = {1, 2, 3} and Y= {abc}.

Thus, the inverse of f exists and f - 1 = g.

f - 1: {abc} →  {1, 2, 3} is given by,

f - 1(a) = 1, f - 1(b) = 2, f-1(c) = 3

Let us now find the inverse of f - 1 i.e., find the inverse of g.

If we define h: {1, 2, 3}  →  {abc} as

h(1) = ah(2) = bh(3) = c, then we have:

, where X = {1, 2, 3} and Y = {abc}.

Thus, the inverse of g exists and g - 1 = h ⇒ (f - 1) - 1 = h.

It can be noted that h = f.

Hence, (f - 1) - 1 = f.

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