Day 8: mat385

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Today:

• Announcements:
  • Your first exam is this weekend.
    • You may take it any time from 5:00 pm today to 8:00 pm on Sunday. It should take about an hour.

    • You'll take it on Canvas (quizzes) using lockdown browser. You will need a web cam.

    • You'll be tested over the five sections, 1.1-1.4 and 2.1

    • You may have a one-sided page of notes on an 8.5x11 sheet of paper.

    • Previous Exam 1s from other years (these may cover more or less material -- and typically we had more time -- 75 minutes):?

      Here are some first exams from years past.

  • I have graded your fifth assignment (section 2.1): problems 4, 20, 32

    • #4: provide counterexamples:
      1. If $a$ and $b$ are integers where $a|b$ and $b|a$, then $a=b$. (1 and -1)
      2. If $n^2>0$ then $n>0$. (-1)
      3. If $n$ is an even integer, then $n^2+1$ is prime. (Wouldn't that be lovely! We could manufacture bigger and bigger primes easily! But it's false: $n=8$. Diamond mentioned $n=0$ -- a very good choice. It brings up the point that 1 is not a prime number. If it were, it would ruin the prime factorization theorem.)
      4. If $n$ is a positive integer, then $n^3>n!$. (How comical! Factorials grow so much faster than powers... $n=1$ gives equality; $n=6$ is the first positive integer a factorial greater than the corresponding cube.)

    • #20: the sum of an integer and its square is even.

      This is one that we might do by exhaustion (cases). There are only two cases: either the integer (call it $n$) is even or odd.

      Case 1: $n$ is even; i.e. $n=2m$ for some integer $m$. Then \[ n+n^2=2m+(2m)^2=2(m+2m^2) \] By the closure of the integers under addition and multiplication, $m+2m^2$ is an integer; hence the sum $n+n^2$ is a product of 2 and an integer, hence even.

      Case 2: $n$ is odd; i.e. $n=2m+1$ for some integer $m$. Then \[ n+n^2=2m+1+(2m+1)^2=2m+1+4m^2+4m+1=2(m+2m^2+2m+1) \] By the closure of the integers under addition and multiplication, $m+2m^2+2m+1$ is an integer; hence the sum $n+n^2$ is a product of 2 and an integer, hence even.

      Some of you simply invoked the result of #19 -- well done! You had already shown that the product of consecutive integers is even, and $m+m^2=m(m+1)$, the product of consecutive integers.

      Make sure that you cite #19, if you make this argument. It's like creating a new rule to use in a proof sequence (mp, ds, cont, #19....). This is why mathematicians have so many theorems lying around. We use them in arguments.

    • #32: if the product of two integers is not divisible by an integer $n$, then neither integer is divisible by $n$.

      This one is a classic for contraposition. Notice how everything is stated in the negative: we might write this as

      ($pq$ divisible by $n$)' $\rightarrow$ (($p$ divisible by $n$)' $\land$ ($q$ divisible by $n$)')

      which is equivalent to

      ($pq$ divisible by $n$)' $\rightarrow$ (($p$ divisible by $n$) $\lor$ ($q$ divisible by $n$))'

      by De Morgan; and finally as

      (($p$ divisible by $n$) $\lor$ ($q$ divisible by $n$)) $\rightarrow$ ($pq$ divisible by $n$)

      by contraposition. Now isn't that a lot easier to prove?

      Important note: To finish off you can exploit symmetry: you can say "without loss of generality (WLOG), assume that $p$ is divisible by $n$", and proceed to the consequent.
      Then there's no need to go back and say it again for $q$. The only difference in the argument would be the letter! So you only have to do one-half of the antecedent. Symmetry is a very powerful force in mathematics, physics, biology (you're bilaterally symmetric), etc.

    • General comments:
      • I didn't grade 10 or 15, but I did notice people trying to prove them "by example". You can't prove general rules by looking at just a few examples. You might have gotten lucky, and picked just the right examples, the ones which work....
        So on #20 (above), for example: you can't just show that it works for the first five integers and say "Therefore,...." It isn't a bad idea to check to make sure that the pattern is working, however: \[ \begin{array}{c} {1+1=2}\cr {2+4=6}\cr {3+9=12}\cr {4+16=20}\cr {5+25=30} \end{array} \] But that's not a proof. You'll have to do an infinity of more cases -- good luck! You've only got an hour for the test...:)

• Today: we'll do a little review:
  • I didn't receive any requests for specific problems, other than a request for #32, which happened to be graded (see above).

  • Here are some 1.3 examples, that a student and I went over during office hours.

  • Here's a general summary of things we've been doing.

  • You can expect the exam to look a lot like the homework problems.

  • I do have an office hour today, from 9:00-10:00, and you're welcome to attend that.

    I'll probably be available at other times, if that doesn't work out (although I have something from 10-11, and will be busy from 4:00 on).

  • Good luck!

• Links:
  • Nothing new to report.