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\flushleft \underline{Homework 1 -- MATH 2510 Spring 2019}\\
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\textbf{\underline{Definitions}} \\
Recall... \\
\textbf{Element of}: To write $a \in A$ means that ``$a$" is an object inside of the set $A$. \\
\textbf{Subset}: $A \subset B$ means for all $a \in A$, it is also true that $a \in B$. \\
\textbf{Cartesian product}: $A \times B = \{(a,b) \colon a \in A, b \in B\}$  \\
\textbf{Relation}: A relation $R$ on a set $X$ is a subset of $X \times X$, i.e. $R \subseteq X \times X$. \\
\textbf{Function}: A function is a relation $R$ where given any two $(a,b) \in R$ and $(c,d) \in R$, it's always the case that $a \neq c$. \\
\textbf{AND ($\wedge$)}: $P \wedge Q$ is true only when both $P$ is true and $Q$ is true \\
\textbf{OR ($\vee$)}: $P \vee Q$ is true whenever either $P$ is true, $Q$ is true, or both $P$ and $Q$ are true \\
\textbf{IMPLIES ($\longrightarrow$)}: $P \rightarrow Q$ is true when both $P$ and $Q$ are true or anytime when $P$ is false \\
\textbf{IF AND ONLY IF ($\longleftrightarrow$)}: $P \leftrightarrow Q$ is true if both $P$ and $Q$ are true or if both $P$ and $Q$ are false \\
\textbf{NEGATION ($\neg$)}: $\neg P$ is true only when $P$ is false \\
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\underline{\textbf{Problems}} \\
\begin{enumerate}
\item Let $X=\{1,3,4\}$, $Y=\{2,4,5\}$, and $Z=\{1,2,3,5,7\}$. 

\begin{enumerate}
\item Is $2 \in X$? Is $2 \in Y$? Is $2 \in Z$?

\item Is $X \subseteq Y$? Is $X \subseteq Z$? Is $Y \subseteq Z$? Is $Z \subseteq X$? Is $Z \subseteq Y$?

\item  What is $X \cup Y$? What is $X \cup Z$? What is $Y \cup Z$? What is $X \cup Y \cup Z$?

\item What is $X \cap Y$? What is $X \cap Z$? What is $Y \cap Z$? What is $X \cap Y \cap Z$? 

\item What is $X \times Y$?

\end{enumerate}

\item Let $X=\{1,3,8\}$. Let $R \subset X \times X$ be a relation given by $R = \{(1,1), (3,1), (8,1) \}$. Is $R$ a function? Why or why not?

\item What does the symbol $\emptyset$ denote in set theory?

\item Is the following string of symbols a formula of propositional logic? Why or why not?
$$(A \vee \neg) \vee (A \wedge (B \wedge \neg A)).$$

\item Is the following string of symbols a formula of propositional logic? Why or why not?
$$(A \wedge (\neg B)) \wedge ((\neg A) \wedge B)$$

\item Let $P$, $Q$, and $R$ be statements. Given that $P$ is true, $Q$ is false, and $R$ is also false, state whether the following formulas are true or false and explain your answer.
\begin{enumerate}[(a)]
\item $P \rightarrow (Q \vee R)$ 
\item $R \rightarrow (Q \wedge P)$ 
\item $\neg P \longleftrightarrow Q$
\item $(P\vee R) \leftrightarrow (Q \vee R)$
\item $P \wedge (Q \wedge (R \vee P))$
\end{enumerate}
\item Make truth tables for the formulas (a), (c), and (e) of Problem 6.
\end{enumerate}
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