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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/00%3A_Front_Matter/04%3A_ColophonEdition 3rd Edition - version 8 ©2021 Al Doerr, Ken Levasseur Applied Discrete Structures by Alan Doerr and Kenneth Levasseur is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike ...Edition 3rd Edition - version 8 ©2021 Al Doerr, Ken Levasseur Applied Discrete Structures by Alan Doerr and Kenneth Levasseur is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. You are free to Share: copy and redistribute the material in any medium or format; Adapt: remix, transform, and build upon the material. You may not use the material for commercial purposes. The licensor cannot revoke these freedoms as long as you follow the license terms.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/10%3A_TreesIn this chapter we will study the class of graphs called trees. Trees are frequently used in both mathematics and the sciences. Our solution of Example 2.1.1 is one simple instance. Since they are oft...In this chapter we will study the class of graphs called trees. Trees are frequently used in both mathematics and the sciences. Our solution of Example 2.1.1 is one simple instance. Since they are often used to illustrate or prove other concepts, a poor understanding of trees can be a serious handicap. For this reason, our ultimate goals are to: (1) define the various common types of trees, (2) identify some basic properties of trees, and (3) discuss some of the common applications of trees.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/11%3A_Algebraic_StructuresThe primary goal of this chapter is to make the reader aware of what an algebraic system is and how algebraic systems can be studied at different levels of abstraction. After describing the concrete, ...The primary goal of this chapter is to make the reader aware of what an algebraic system is and how algebraic systems can be studied at different levels of abstraction. After describing the concrete, axiomatic, and universal levels, we will introduce one of the most important algebraic systems at the axiomatic level, the group. We will close the chapter with a discussion of how some computer hardware and software systems use the concept of an algebraic system.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/12%3A_Boolean_Algebra/12.01%3A_Posets_RevisitedConsider the poset \((\mathcal{P}(A),\subseteq)\text{,}\) where \(A = \{1, 2, 3\}\text{.}\) The greatest lower bound of \(\{1, 2\}\) and \(\{1,3\}\) is \(\ell = \{1\}\text{.}\) For any other element \...Consider the poset \((\mathcal{P}(A),\subseteq)\text{,}\) where \(A = \{1, 2, 3\}\text{.}\) The greatest lower bound of \(\{1, 2\}\) and \(\{1,3\}\) is \(\ell = \{1\}\text{.}\) For any other element \(\ell'\) which is a subset of \(\{a, b\}\) and \(\{a, c\}\) (there is only one; what is it?), \(\ell' \subseteq \ell\text{.}\) The least element of \(\mathcal{P}(A)\) is \(\emptyset\) and the greatest element is \(A=\{a, b, c\}\text{.}\) The Hasse diagram of this poset is shown in Figure \(\PageInd…
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/17%3A_Appendix/17.03%3A_C_-_DeterminantsIn Chapter 5 we defined the determinant of a \(2 \times 2\) matrix for the sole purpose of providing some hands-on experience in the computation of inverses of \(2 \times 2\) matrices. In this appendi...In Chapter 5 we defined the determinant of a \(2 \times 2\) matrix for the sole purpose of providing some hands-on experience in the computation of inverses of \(2 \times 2\) matrices. In this appendix we will define the determinant of any square matrix, and summarize the main properties of determinants.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/00%3A_Front_Matter/01%3A_TitlePageApplied Discrete Structures Al Doerr Ken Levasseur Department of Mathematical Sciences University of Massachusetts Lowell kenneth_levasseur@uml.edu May 25, 2021
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/00%3A_Front_Matter/07%3A_PrefaceWe had signed a contract for the second edition with Science Research Associates in 1988 but by the time the book was ready to print, SRA had been sold to MacMillan. The open source software movement ...We had signed a contract for the second edition with Science Research Associates in 1988 but by the time the book was ready to print, SRA had been sold to MacMillan. The open source software movement was just starting in the late 1980's and in 2005, the first version of Sage (later renamed SageMath), an open-source computer algebra system, was first released.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/04%3A_More_on_SetsIn basic algebra we are aware that \(a \cdot (b + c) = a\cdot b + a \cdot c\) for all real numbers \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) In logic we verified an analogue of this statement, nam...In basic algebra we are aware that \(a \cdot (b + c) = a\cdot b + a \cdot c\) for all real numbers \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\) In logic we verified an analogue of this statement, namely, \(p \land ( q \lor r) \Leftrightarrow (p \land q)\lor (p \land r))\text{,}\) where \(p, q, \textrm{ and } r\) were arbitrary propositions.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/12%3A_More_Matrix_Algebra/12.05%3A_Some_Applications\begin{equation*} \begin{split} e^Ae^B &= \left(\sum _{k=0}^{\infty } \frac{A^k}{k!}\right) \left(\sum _{k=0}^{\infty } \frac{B^k}{k!}\right)\\ & =\left(I + A+\frac{A^2}{2}+ \frac{A^3}{6}+ \cdots \rig...\begin{equation*} \begin{split} e^Ae^B &= \left(\sum _{k=0}^{\infty } \frac{A^k}{k!}\right) \left(\sum _{k=0}^{\infty } \frac{B^k}{k!}\right)\\ & =\left(I + A+\frac{A^2}{2}+ \frac{A^3}{6}+ \cdots \right)\left(I +B+\frac{B^2}{2}+ \frac{B^3}{6}+ \cdots \right)\\ &= I + A + B+ \frac{A^2}{2}+ A B + \frac{B^2}{2}+\frac{A^3}{6}+ \frac{A^2B}{2}+\frac{A B^2}{2}+ \frac{B^3}{6}+\cdots \\ &= I + (A+B) + \frac{1}{2}\left(A^2+ 2 A B + B^2\right)+ \frac{1}{6}\left(A^3+ 3A^2B+ 3A B^2+ B^3\right)+\cdots \\ &=I…
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Discrete_Structures_(Doerr_and_Levasseur)/10%3A_Trees/10.02%3A_Spanning_TreesThe method that we have used (in Step 2.1) to select a bridge when more than one minimally weighted bridge exists is to order all bridges alphabetically by the vertex in \(L\) and then, if further tie...The method that we have used (in Step 2.1) to select a bridge when more than one minimally weighted bridge exists is to order all bridges alphabetically by the vertex in \(L\) and then, if further ties exist, by the vertex in \(R\text{.}\) The first vertex in that order is selected in Step 2.1 of the algorithm.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/12%3A_Boolean_Algebra/12.02%3A_Lattices\begin{equation*} \begin{array}{cc} \begin{array}{c|ccccc} \lor & \pmb{0} & a & b & c & \pmb{1} \\ \hline \pmb{0} & \pmb{0} & a & b & c & \pmb{1} \\ a & a & a & \pmb{1} & \pmb{1} & \pmb{1} \\ b & b & ...\begin{equation*} \begin{array}{cc} \begin{array}{c|ccccc} \lor & \pmb{0} & a & b & c & \pmb{1} \\ \hline \pmb{0} & \pmb{0} & a & b & c & \pmb{1} \\ a & a & a & \pmb{1} & \pmb{1} & \pmb{1} \\ b & b & \pmb{1} & b & \pmb{1} & \pmb{1} \\ c & c & \pmb{1} & \pmb{1} & c & \pmb{1} \\ \pmb{1} & \pmb{1} & \pmb{1} & \pmb{1} & \pmb{1} & \pmb{1} \\ \end{array} & \begin{array}{c|ccccc} \land & \pmb{0} & a & b & c & \pmb{1} \\ \hline \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} \\ a & \pmb{0} & …
