Math 199 (Assignments)

[Not to be turned in, but we will discuss in class]

1. Let $\R$ be the set of all real numbers, let $X$ be the set $\{\hbox{Alan},\hbox{Betty},\hbox{Charles},\hbox{Debbie}\}$. Let $P(x)$ be the proposition "$x$ loves Alan", and $Q(y)$ the proposition "Alan loves $y$". Write the following statements using the symbols of propositional logic (where the people involved are the ones in the set $X$). Note that these should all be propositions of the form $R$, not $R(x)$ or $R(a,b)$, so there should be no unquantified variables.
   1. Everybody loves Alan.
   2. Alan loves everybody.
   3. Alan loves nobody.
   4. Nobody loves Alan.
   5. $0\cdot x=0$ for every real number $x$.
   6. $x^2>4$ whenever $x>2$.
   7. If $a > b$, then $ac < bc$ whenever $c < 0$.
2. Suppose you have an English paper, a Math 199 homework, and a Chemsitry lab to do. Each item is either done or not done, so there are eight possibilities for what you have finished so far:

   	English	Math	Chemistry
   1.	Not Done	Not Done	Not Done
   2.	Done	Not Done	Not Done
   3.	Not Done	Done	Not Done
   4.	Done	Done	Not Done
   5.	Not Done	Not Done	Done
   6.	Done	Not Done	Done
   7.	Not Done	Done	Done
   8.	Done	Done	Done

   1. For each of the sentences below, list the situations above that satisfy the statement:
      1. All my work is done
      2. Some of my work is done
      3. None of my work is done
      4. All my work isn't done
      5. Some of my work isn't done
      6. None of my work isn't done
      7. Not all my work is done
      8. Not all my work isn't done
   2. Consider the phrases
      1. "For all my tasks $x$, ..."
      2. "There exists a task $x$ where ..."
      3. "There does not exist a task $x$ where ..."
      4. "It's not the case that for all my tasks $x$, ...".

      Write each of the sentences from part (a) in two distinct ways using two different initial phrases from the list above together with one of the propositions "$x$ is done" or "$x$ is not done". Say something about why you think you are right in each case.

Math 199 (Fall 2019) web pages Created: 25 Sep 2019 Last modified: 25 Sep 2019 at 3:44 PM Comments to: dpvc@union.edu

Assignments