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CSC 165           Homework Assignment 2 -- Marking Scheme       Winter 2009
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NOTE TO STUDENTS:

You will find below the marking scheme used for your homework, including
the meaning of marking codes and number of marks associated with each one.
This file also contains my instructions to the marker (so you can get an
idea of how the homework was marked) and the marker's comments about each
question.

Please take the time to read this carefully before you ask questions about
the grading of your homework.


NOTE TO MARKER:

Be picky!  On any homework, it is the responsibility of students to show
that they understand how to solve each problem and to write up their
answers carefully.  Remember that marking is not only about evaluating a
student's performance, but also mostly about giving them feedback so that
they can learn from their mistakes.  This is especially important for
students who made numerous or more serious mistakes, as they are likely to
need more feedback in order to understand why their answers were incorrect.

For each question, I list solution elements with an associated code for
writing on student papers (the letter(s) between underscores _) and a
number of marks.  There are also general errors (with associated codes)
given below, with a maximum number of marks to take off for each type of
general error (as a percentage of the value of the question).  You will
likely encounter other common errors, or maybe decide to break down the
marking scheme further.  Simply make note of these changes/additions to
the marking scheme, and introduce new code letters (or short words) to
allow you to quickly give accurate feedback to the students (both in terms
of what they did wrong and how many marks it cost them).  Your marking
comments will be included in this marking scheme and posted on the course
website so that students may look up the meaning of marking codes and
understand how their work was marked.


GENERAL ERRORS (marked negatively, in addition to any other errors):

    _N_ notation [up to 20%]:  incorrect/undefined/ambiguous notation

    _V_ vagueness [up to 20%]:  incorrect/unjustified/vague claim


1.  [15 marks]

    For each part:

    _A_ answer [2 marks]:  correct simplified form

    _J_ justification [3 marks]:  correct derivation or good justification
        that does *not* use truth tables (or anything directly equivalent)
        (give 1 if the answer is exactly right but there is no explanation,
         since the question could be read that way)

    Marker's Comments:

     -  If there were no mistakes between steps but the answer could be
        simplified further you received an IC error code.  You still
        received full justification marks.

     -  If your answer was incorrect (due to incorrect algebra somewhere)
        you received no marks for your answer but if the steps done
        correctly were justified you only lost justification marks for the
        step where algebra was performed incorrectly.

     -  Students may have received a general _V_ error if they did not
        state any laws/justifications for particular steps.

    Extra Error Codes:

    _IA_ incorrect algebra:  a step was performed incorrectly
    _IC_ incomplete simplification:  answer could be simplified further


2.  [12 marks]

    For each part:

    _S_ structure [3 marks]:  clear and correct detailed proof structure
        (including working with the correct negation for part (b))

    _C_ content [3 marks]:  correct proof

    Marker's Comments / Error Codes:

     -  Students received full marks for _S_ if the structure of their
        proof made sense for what they were trying to show.  Points were
        lost of parts of the structure did not make sense or for example if
        undefined variables were introduced or assumptions did not make
        sense.

     -  Content marks were lost if mistakes were made within the proof, or
        if the content did not match with the structure of the proof.

    _IN_ incorrect negation (for part b):  negation of (S2) was done
        incorrectly


3.  [11 marks]

    _S_ structure [5 marks]:  clear and correct detailed proof structure

    _C_ content [6 marks]:  correct proof (in particular, correct handling
        of "\exists !")

    Marker's Comments / Error Codes:

     -  If the proof did not show that the value of c is unique, the
        student lost 4 _C_ marks and 3 _S_.  In most cases this meant the
        student did not show both directions of the bi-conditional.

     -  Students who did not prove the statement for any a,b matching the
        constraints also lost significant _C_ marks, i.e., those choosing
        specific values for a or b, or both.


4.  [18 marks]

    For each part:

    _S_ structure [3 marks]:  clear and correct detailed proof structure

    _C_ content [3 marks]:  correct proof

    Marker's Comments / Error Codes:

     -  Marks were lost if strange assumptions were made about x, or if the
        statement being proved was used in the proof itself.


5.  [11 marks]

    _S_ structure [5 marks]:  clear and correct detailed proof structure

    _C_ content [6 marks]:  correct proof

    Marker's Comments / Error Codes:

    _IOE_ incorrect 'even' and 'odd' usage [-3 _S_]:  terms and definitions
        of 'even' and 'odd' applied to rational numbers


6.  [13 marks]

    _C_ conjecture [4 marks]:  correct conjecture

    _S_ structure [5 marks]:  clear and correct detailed proof structure

    _P_ proof [4 marks]:  correct proof

    Marker's Comments:

     -  Regardless of what conjecture you chose, if your proof structure
        and the proof itself were valid for the given conjecture you
        received full marks for _S_ and _P_.

    Error Codes:

    _SC_ simple conjecture [-3 _C_]:  using a conjecture that is an overly
        simple relationship that one could find by looking up the
        definitions of the means