14-240/Tutorial-October7

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Boris

Subtle Errors in Proofs

Check out these proofs:

Proof 1

Let , be subspaces of a vector space . We show that is a subspace

.

Assume that is a subspace.
Let , .
Then .
Then .
Then .
Case 1: :
Since and has additive inverses, then .
Then .
Case 2: :
Since and has additive inverses, then .
Then .
Then .
Then . Q.E.D.
Proof 2

Let . Then , define

and . We show that is not a vector

space over .

We show that is not commutative.
Let .
Then .
Then is not commutative.
Then is not a vector space. Q.E.D.


Can you spot the subtle error in each?


Error in Proof 1

The error in the proof is the logical equivalence of these two statements.

(1) Let . ... Then .

(2) Then .

Error in Proof 2

Nikita