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In Proof 1, the equivalence of the last line and the "let" statement to the second last line is not obvious: |
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In Proof 1, the equivalence of (a) the last line and (b) the "let" statement to the second last line is not obvious: |
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(1) Let <math>x \in W_1, y \in W_2</math>. [Many lines] Then <math>x \in W_2 \or y \in W_1</math>. |
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(1) Let <math>x \in W_1, y \in W_2</math>. [Many lines] Then <math>x \in W_2 \or y \in W_1</math>. |
Revision as of 20:30, 7 November 2014
Welcome to Math 240! (additions to this web site no longer count towards good deed points)
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#
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Week of...
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Notes and Links
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1
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Sep 8
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About This Class, What is this class about? (PDF, HTML), Monday, Wednesday
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2
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Sep 15
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HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf
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3
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Sep 22
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HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf
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4
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Sep 29
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HW3, Wednesday, Tutorial, HW3_solutions.pdf
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5
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Oct 6
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HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf
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6
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Oct 13
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No Monday class (Thanksgiving), Wednesday, Tutorial
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7
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Oct 20
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HW5, Term Test at tutorials on Tuesday, Wednesday
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8
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Oct 27
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HW6, Monday, Why LinAlg?, Wednesday, Tutorial
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9
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Nov 3
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Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial
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10
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Nov 10
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HW8, Monday, Tutorial
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11
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Nov 17
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Monday-Tuesday is UofT November break
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12
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Nov 24
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HW9
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13
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Dec 1
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Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial
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F
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Dec 8
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The Final Exam
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Register of Good Deeds
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![Class Photo](/images/thumb/6/6b/14-240-ClassPhoto.jpg/310px-14-240-ClassPhoto.jpg) Add your name / see who's in!
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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?
In Proof 1, the equivalence of (a) the last line and (b) 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 or into proof in which you assume that one of the conditions in the disjunction is satisfied.
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.
Nikita