Discrete Math: A proof follows the Theorem below. Is it a valid proof or is ther

Question

Discrete Math:

A proof follows the Theorem below. Is it a valid proof or is there na error in the proof, and if so, what is the error?

Theorem: The sum of two rational numbers is rational.

"Proof: Suppose r and s are rational numbers. Then r = a/b and s = c/d for some integers a, b, c, and d with b 0 and d 0 (by defintion of rational).

Then r + s = a/b + c/d (by substitution)= ad + bc/bd ( by basic algebra)
Let P=ad+bc and q=bd. Then p and q are integers because products and sums of interger are integers and because a,b,c and d are all integers.

Also, q 0, by the zero product property.

Thus r + s = p/q where p and q are integers and q 0.

Therefore, r + s is rational by definition of a rational number

A. Valid Proof

B. The author of the oriif was arguing from examples and thus the proof has error.

C. The proof exhibits circular reasoning and thus has an error.

D.The author of the proof was confused between what was known and what was to be shown and thus the proof has an error.

theorem. ts k a vald proof or is there an error in thep e an error in the proot, and if so, what is the error? Theorem: the sum of two rational numbers is rational. Proof: Proof: Suppose r and s are rabonal numbers. Then-albands. cid for some Suppose r and s are rational numbers. Then r a/b and s c/d for some integers a, b, c, and d with bo and d+o (by definition of rational]. Then r +by substitution by substitution Then r + s . adtbc by basic algebra. bd . ad·he and. bd. Then, ind a are integers because products and s sums of integers are integers and because a, b, c, and d are all integers. Also q a 0 by the zero p e , b, c, and d are all integers. Also q o by the zere product iet property Thus, rs whre pand q are integers and q o Therefore, r+ s is rational by definition of a rationsl number.

Explanation / Answer

Hi,

The proof given is called direct proof and is correct, so the answer is A.valid proof.

lets take a moment to understand what can go wrong in such proofs,

1.Arguing from examples: this is common, this is a wrong practice because a proof should always be generic and not example specific.

2.Using same letter to mean two different things- this is another mistake that can occur where same variable is used for 2 diff things which is essentially wrong.

3.Jumping to a conclusion- this is another mistake where we jump to a conclusion with out any adequate reasons

Thumbs up if this was helpful, otherwise let me know in comments. Good Day.

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