Consider the following theorem:The function f :  →  defined by f(x) = 5x + 3 is injective....

Consider the following theorem:

The function f :  →  defined by f(x) = 5x + 3 is injective.

Indicate what, if anything, is wrong with each of the following "proofs."

(a) Let x1. x2 ∈  and suppose x1 = x2. Then 5x1 = 5x2 and 5x1 + 3 = 5x2 + 3, so f(x1) = f(x2). Thus f is injective.

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(b) Let x1, x2 ∈  and suppose f(x1) = f(x2). We must prove that x1 = x2. Now .f(x1) = 5x1 + 3 and .f(x2) = 5x2 + 3. Since x1 = x2, we have 5x1 + 3 = 5x2 + 3. It follows that 5x1 = 5x2 and x1 = x2. Thus f is injective.

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(c) Let x1, x2 ∈  and suppose x1 ≠ x2. Then 5x1 ≠ 5x2 and 5x1 + 3 ≠ 5x2 + 3, so f(x1) ≠f(x2). It follows that x1 = x2 whenever f(x1) = f(x2), and f is injective.

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(d) Let x1, x2 ∈  and suppose f(x1) ≠ f(x2). Thus 5x1 + 3 ≠  5x2 + 3 and 5x1 ≠ 5x2, so x1 ≠ x2. It follows that f(x1) = f (x2) only if x1 = x2, and f is injective.

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(e) We have f(1) = 8 and f(2) = 13 , so if x1 ≠ x2, then f(x1) ≠ f(xz). It follows that f is injective.

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(f) Let x1, x2 ∈  and suppose f(x1) = f(x2). Then 5x1 + 3 = 5x2 + 3 and 5x1 = 5x2, so x1 = x2. Thus f is injective.