# dcernst/IBL-IntroToProof

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- modified some wording
- adding a couple of exercises
 @@ -25,8 +25,8 @@ \pagestyle{fancy} -\lhead{\scriptsize Course Notes for Logic, Proof, \& Axiomatic Systems (Spring 2011)} -\rhead{\scriptsize Instructor: \href{http://oz.plymouth.edu/~dcernst}{D.C. Ernst}} +\lhead{\scriptsize Course Notes for Introduction to Proof (Version 1.1)} +\rhead{\scriptsize Instructor: \href{http://danaernst.com}{D.C. Ernst}} \lfoot{\scriptsize This work is an adaptation of notes written by Stan Yoshinobu of Cal Poly and Matthew Jones of California State University, Dominguez Hills.} \cfoot{} \renewcommand{\headrulewidth}{0.4pt} @@ -36,6 +36,7 @@ \newtheorem{theorem}{Theorem}[section] \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} +\newtheorem{question}[theorem]{Question} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} @@ -60,18 +61,16 @@ \addtocounter{section}{0} -\begin{section}{Introduction to Mathematics} +\begin{section}{Introduction to Mathematics (Continued)} \addtocounter{subsection}{3} -\addtocounter{theorem}{54} +\addtocounter{theorem}{50} %%%% Quantification \begin{subsection}{Introduction to Quantification} -Quantification is often where students struggle in higher mathematics. Not only are the concepts abstract, the concepts must be carefully delineated into cases. - -Recall that sentences of the form $x>0$" are not propositions (unless the context of $x$ is perfectly clear). In this case, we call $x$ a \textbf{free variable}. +Recall that sentences of the form $x>0$" are not propositions (unless the context of $x$ is perfectly clear). In this case, we call $x$ a \textbf{free variable}. In order to turn a sentence with free variables into a proposition, for each free variable, we need to either substitute in a value (not necessarily a number) for the free variable or we must quantify" the free variable. \begin{definition} A sentence with a free variable is called a \textbf{predicate}. @@ -81,9 +80,16 @@ Give 3 examples of mathematical predicates involving 1, 2, and 3 free variables, respectively. \end{exercise} -It is convenient to borrow function notation to represent predicates. For example: $S(x):=x^2-4=0"$, $L(a,b):=a0$". Is this proposition true or false? The answer depends on what $x$'s we are taking \emph{all} of. For example, if the universe of discourse is the set of integers, then the statement is false. However, if we take the universe of acceptable values to be the natural numbers, then the proposition is true. We must be careful to avoid such ambiguities. Often, the context can resolve such ambiguities, but otherwise, we need to write things like: For all $x\in\mathbb{Z}$, $x>0$" or For all $x\in\mathbb{N}$, $x>0$". @@ -127,6 +133,14 @@ If a predicate has more than one free variable, then we can build propositions by quantifying each variable. However, the order of the quantifiers is extremely important!!! +\begin{exercise}\label{exer:ways to quantify} +Let $P(x,y)$ be a predicate with the free variables $x$ and $y$ (and let's assume the universe of discourse is clear). Write down all possible ways (where order matters) that the variables could be quantified. To get you started, here's one: For all $x$, there exists $y$ such that $P(x,y)$. Find the rest. +\end{exercise} + +\begin{problem} +Are there any propositions on your list from Exercise~\ref{exer:ways to quantify} that are equivalent to others on your list? +\end{problem} + \begin{exercise} Suppose that the universe of acceptable values is the set of married people. Consider the predicate $M(x,y):=x\mbox{ is married to }y"$. Discuss the meaning of each of the following. \begin{enumerate} @@ -147,8 +161,12 @@ Repeat the exercise above but replace the existential quantifiers with universal quantifiers. \end{exercise} +\begin{problem} +Conjecture a summary of the various possibilities for quantifying predicates involving two variables. You do \emph{not} need to prove your conjecture. +\end{problem} + \begin{exercise} -Suppose that the universe of acceptable values is the set of real numbers. Consider the predicate $G(x,y):=x>y"$. There are six ways to bind the variables of this predicate (why six?). Find all six propositions and determine the truth value of each (you do not need to prove your answers). Do any of the six have the same meaning as one of the others? +Suppose that the universe of acceptable values is the set of real numbers. Consider the predicate $G(x,y):=x>y"$. Find all possible \emph{distinct} ways to bind the variables to create propositions and then determine the truth value of each (you do not need to prove your answers). Do any of your propositions have the same meaning as one of the others? \end{exercise} \end{subsection}