Question

Sometimes wrong proofs are written claiming that a statement is true. They are called bogus proofs. Explain what is wrong in the following bogus proof. If a|b and b|a, then a = b. Bogus proof: Let a and b be integers such that a|b and b|a. Then by definition, b = ar for some integer r and a = bs for some integer s. Therefore, a = ars (by substitution), which implies 1 = rs. Since r and s are integers, both r and s must be 1 for rs to be 1. So, r = 1 and s = 1, which implies a = b. Hence if a|b and b|a, then a = b.

          Sometimes wrong proofs are written claiming that a statement is true. They are called bogus proofs. Explain what is wrong in the following bogus proof.

If a|b and b|a, then a = b.

Bogus proof:

Let a and b be integers such that a|b and b|a.

Then by definition, b = ar for some integer r

and a = bs for some integer s.

Therefore, a = ars (by substitution), which implies 1 = rs.

Since r and s are integers, both r and s must be 1 for rs to be 1.

So, r = 1 and s = 1, which implies a = b.

Hence if a|b and b|a, then a = b.

Sometimes wrong proofs are written claiming that a statement is true. They are called bogus proofs. Explain what is wrong in the following bogus proof.

If a|b and b|a, then a = b.

Bogus proof:

Let a and b be integers such that a|b and b|a.

Then by definition, b = ar for some integer r

and a = bs for some integer s.

Therefore, a = ars (by substitution), which implies 1 = rs.

Since r and s are integers, both r and s must be 1 for rs to be 1.

So, r = 1 and s = 1, which implies a = b.

Hence if a|b and b|a, then a = b.

Added by Timothy P.

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