14240/TutorialOctober7

Contents 
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?
In Proof 1, the equivalence of (2) the last line and (1) the "let" statement to the second last line is not obvious:
(1) Let . [Many lines] Then .
(2) Then .
Rewrite sentences (1) and (2) into a form that is easier to compare:
(1) .
(2) .
For Proof 1 to be correct, we must show that sentences (1) and (2) are equivalent. Alternatively, alter the structure of Proof 1 into a proof by contradiction.
In Proof 2, the only thing that is shown is that is not the additive identity. For Proof 2 to be correct, either plug in a vector that is not or show that is the additive identity by some other means, which introduces a contradiction.