If you want to test your propositional logic skills, try this puzzle that arose from a conversation at my office hours:

Suppose we are given the premises (1) $p\to q$ and (2) $\neg p\to q$.

a) What can we infer from (1) and (2)?

b) The contrapositive of (1) is $\neg q\to\neg p$. Putting that together with (2), we can infer $\neg q\to q$. Is that a contradiction? Why or why not?

Answers below…

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Graded: 4.2 #32, 4.4 #8

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The first midterm will be this Friday, July 11, from 2:00 to 3:00 in 4 Evans. We’ll start as close to 2:00 as possible, not at 2:10.

The only items you will need (and the only items that are allowed) are pens/pencils, erasers, and pencil sharpeners. You don’t need to bring a blue book; paper will be provided. No notes, books, or electronics may be used during the test.

This midterm will cover sections 1.1, 1.3-1.8, 2.1-2.5, and 4.1-4.4. The second hour of class on Thursday is reserved for review, and I encourage you to visit office hours if you want additional help. (The schedule is on the syllabus page.)

Sample midterms:

• Prof. Williams’ Spring 2014 sample midterm (solutions).
• Prof. Haiman’s Fall 2012 test (solutions). Problem #5 is out of bounds.
• Prof. Sturmfels’ Spring 2012 test (solutions). Problems #3, 4 are out of bounds.
• Prof. Sturmfels’ Spring 2010 test (solutions). Problems #1, 3 are out of bounds.

The timing and selection of topics in Math 55 has varied from term to term; problems listed as “out of bounds” above would not appear on our first midterm. In general, problems on the midterm will resemble problems that have been assigned as homework.

Graded: 4.1 #16, Supplemental Problem #4.

(Note: 4.1 #8 is not being graded, but if you believed you had a proof of what turns out to be a false statement, then it is a very worthwhile exercise to figure out which step of your proof contains a logical error. There is guaranteed to be one!)

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Graded: 2.1 #38, 2.3 #40

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These problems are part of the homework due on Monday, July 7. (Note: I made a minor correction to #2 a couple hours after the initial posting.)

1. For each integer $n\ge 1$, let $A_n$ be the set defined by $$A_n = \{x\in\mathbb R ~|~ n^2 \le x \le (n+10)^2\}.$$ Determine the sets $\bigcup\limits_{n=1}^\infty A_n$ and $\bigcap\limits_{n=1}^\infty A_n$. Prove your answers.
2. Let $f:\mathbb R-\{-1\}\to\mathbb R$ be defined by $f(x)=\dfrac x{x+1}$. Determine the range of $f$ and prove your answer.
3. Let $F:\mathbb R\times\mathbb R\to\mathbb R\times\mathbb R$ be defined by $F(x,y)=(x+y,x-y)$. Show that $F$ is a bijection, and determine its inverse function.
4. Let $S$ be the set of infinite sequences of integers. For example, the entire Fibonacci sequence $$(1,1,2,3,5,8,13,\ldots)$$ is one element of $S$. Show that $S$ is not countable. (Hint: Use an argument similar to the proof that $\mathbb R$ is uncountable.)