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    Preface%
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\noindent 
Unlike in earlier courses, success in upper-level undergraduate mathematics classes (and beyond) does not depend nearly so much on being able to find the right answer to a question as it does on being able to provide a convincing explanation that the answer is correct. (Mathematicians call this explanation a \define{proof}.)
This textbook %for Math~2000 (``Mathematical Concepts''), a course at the University of Lethbridge that 
is designed to help students acquire this essential skill, by developing a working knowledge of:
	\begin{enumerate}
	\item proof techniques (and their basis in Logic),
	and
	\item fundamental concepts of abstract mathematics.
	% (such as sets, functions, %equivalence relations, and cardinality).
	\end{enumerate}


We start with the language of {\SL}, where the rules for proofs are very straightforward. Adding sets and quantifiers to this yields First-Order Logic, which is the language of modern mathematics. Being able to do proofs in this setting is the main skill necessary for success in advanced classes in pure mathematics (and in mathematics research).

It is also important to be familiar with (and be able to prove statements about) sets and functions, which are the building blocks of all mathematical objects. In addition, a \lcnamecref{CardinalityChap} on \emph{Cardinality} provides an introduction to the surprising notion of ``uncountable sets'': infinite sets with so many elements that it is impossible to make a list $x_1,x_2,x_3,\ldots$ of all of them (even if the list is allowed to be infinitely long).