Consider the following statement. The sum of any two rational numbers is a rational number. The statement is true, but the following proposed proof is incorrect. Proposed proof: 1. Suppose r and s are any rational numbers. a 2. By definition of rational,r =- and s = for some integers a, b, c, and d with b * 0 and d # 0. a 3. By substitution, r +s = 9. 4. Thus r +s is a sum of two fractions, which is a fraction. 5. So r +s is a rational number since a rational number is a fraction. Identify the error(s) in the proposed proof. (Select all that apply.) The second sentence should say b = 0 and d = 0 instead of b # 0 and d # 0. The first sentence claims that r and s are rational numbers, which is equivalent to assuming what is to be proved. To prove the statement, r and s must have the same denominator. The fourth sentence assumes that a sum of two fractions is a fraction, which is equivalent to assuming what is to be proved. The second sentence should say a # 0 and c # 0 instead of b # 0 and d # 0.

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