Consider the following statement. The sum of any two rational numbers is a rational number. The statement is true, but the

Question

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. Supposer and s are any rational numbers.
2. By definition of rational, r =
a
b
C
and s = for some integers a, b, c,
d
and d with b #0 and d# 0.
a
C
3. By substitution, r + s = +
b
4. Thus r + s is a sum of two fractions, which is a fraction.
5. Sor+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 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.
The second sentence should say b = 0 and d = 0 instead of b = 0 and d = 0.

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