glossary.tex

\section*{Mathy Terms}
\begin{definition}
\textbf{abstract algebra.}  Abstract algebra is the study of algebraic structures such as groups, rings, and fields.  A course in abstract algebra should be taken following linear algebra.      
\end{definition}
\begin{definition}
  \textbf{binomial coefficient.}  The binomial coefficient $\binom{n}{k}$ is a useful combinatorial tool defined by the factorial formula $\frac{n!}{k!(n - k)!}$ that is valued for its symmetric properties.  
\end{definition}
\begin{definition}
\textbf{bijection.}  A bijection is a function that is one-to-one and onto.
\end{definition}
\begin{definition}
\textbf{calculus.}  Calculus is an early college course that focuses on continuous functions.     
\end{definition}
\begin{definition}
 \textbf{combination.} A combination is a selection of items from a set where the order of elements does not matter.  Hence a combination differs from a permutation.     
\end{definition}
\begin{definition}
    \textbf{factorial.}  A factorial $n!$ is an operation that multiplies a positive integer $n$ by all positive integers less than or equal to it.  
\end{definition}
\begin{definition}
 \textbf{function.}  A function is a mapping of elements from one set $X$ to another $Y$  such that each element of $X$ is mapped to exactly one element of $Y$.  One type of function is a bijective function.  
\end{definition}
\begin{definition}
\textbf{integer.}  Integers can be positive or negative whole numbers (and include 0).  
\end{definition}
\begin{definition}
 \textbf{linear algebra.}  Linear algebra is the study of vector spaces, linear transformations, and solving linear equations.  One should take a college course in linear algebra prior to one in abstract algebra.     
\end{definition}
\begin{definition}
  \textbf{one-to-one.}  A function is said to be one-to-one if it maps each element of the first set maps to at most one element of the second set.   
\end{definition}
\begin{definition}
  \textbf{onto.}  A function is said to be onto if for every element in the second set, there exists at least one element in the first set which outputs it when the function is applied.   
\end{definition}
\begin{definition}
\textbf{permutation.}  A permutation is a bijection from a set to itself that rearranges elements of the set.  The collection of all permutations of a set is the symmetric group $S_n$.  
\end{definition}
\begin{definition}
 \textbf{set.}  A set is a collection of objects we refer to as elements.  
\end{definition}
\begin{definition}
 \textbf{symmetric group $S_n$.}  The symmetric group $S_n$ defined over any set is the group whose elements are all the bijections (i.e. permutations) from the set to itself.   We compute the number of elements in $S_n$ by the factorial $n!$.  
\end{definition}
\begin{definition}
    \textbf{symmetric matrix.}
The entries of a symmetric matrix must be symmetric with respect to the main diagonal.  If you were to flip the matrix along its diagonal (hence switching the rows and columns of the matrix) the flipped matrix would be be the same - invariant under the action of flipping.         
\end{definition}
\begin{definition}
   \textbf{vector space.}  A vector space is a set whose elements are vectors that can be added together and multiplied by certain numbers known as scalars.  Vector spaces are fundamental to the subject of linear algebra.  
\end{definition}