From ecroot@math.gatech.edu Mon Feb  6 11:40:16 2012
Date: Mon, 6 Feb 2012 11:40:15 -0500 (EST)
From: Ernie Croot <ecroot@math.gatech.edu>
To: "Zhu, Zhenchao" <zzhu43@gatech.edu>
Subject: Re: hello professor



Dear Zhenchao,

I put the homeworks in your mailbox.  Grade the following 3 problems
for accuracy (10 points each) [answers may vary]:

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#1.  a.  False.
      b.  False.
      c.  False.
      d.  True.
      e.  False.
      f.  False.

#5.  Let x in A cap C.  This is logically equivalent to saying that
(x in A) and (x in C) [by the definition of set intersection].
Since A subset of B is equivalent to (x in A ==> x in B),
and since C subset of D is equivalent to (x in C ==> x in D), we
have that

(x in A) and (x in C) ==> (x in B) and (x in D) <==> (x in B cap D).

And so, since x was arbitrary we have that (A cap C) is a subset of
(B cap D).

#6.  Let D = B union C.  Then, the complement of (A union B union C)
= the complement of (A union D).  By de Morgan's law for two sets
we conclude that

complement of (A union D) = (complement of A) cap (complement of D)
                           = (comp. of A) cap (comp. of (B union C)).

By de Morgan again, applied to that last union, we conclude that this is
                           = (comp. of A) cap ((comp of B) cap (comp. C)).

Removing the ()'s we are done.

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Grade the rest for effort out of a total of 70 points (total HW score
= accuracy+effort = 30+70 = 100).  Aim for a median score of 90 or higher.

Best regards,

Ernie