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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/zz%3A_Back_Matter/20%3A_GlossaryExample and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pag...Example and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] (Optional) Caption for Image (Optional) External or Internal Link (Optional) Source for Definition "Genetic, Hereditary, DNA ...") (Eg. "Relating to genes or heredity") The infamous double helix CC-BY-SA; Delmar Larsen Glossary Entries Definition Image Sample Word 1 Sample Definition 1
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/12%3A_Cardinality/12.01%3A_Finite_SetsFor m∈N we have defined the counting set N<m={n∈N|n<m}={0,1,…,m−1}.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/18%3A_Equivalence_relations/18.01%3A_MotivationThere are often situations where we want to group certain elements of a set together as being “the same.”
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/17%3A_Relations/17.05%3A_ActivitiesDraw the directed graph for \(\vert\) on the set \(A = \{2,4,6,8,10,12,14,16\}\text{.}\) Then describe how to obtain the graph for the symmetric relation \(\vert \cup \vert ^{-1}\) as an undirected gr...Draw the directed graph for \(\vert\) on the set \(A = \{2,4,6,8,10,12,14,16\}\text{.}\) Then describe how to obtain the graph for the symmetric relation \(\vert \cup \vert ^{-1}\) as an undirected graph from the graph of \(R\) using only an eraser.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/14%3A_Graphs/14.05%3A_ActivitiesEvery pair of a subset \(V' \subseteq V\) and a subcollection \(E' \subseteq E\) defines a subgraph \(G' = (V',E')\) of \(G\text{.}\) Additionally, draw an edge between a member of your group and anot...Every pair of a subset \(V' \subseteq V\) and a subcollection \(E' \subseteq E\) defines a subgraph \(G' = (V',E')\) of \(G\text{.}\) Additionally, draw an edge between a member of your group and another student if that pair was in a group together last class. For each of the following graphs, write out its formal definition as either a (regular) graph, a weigthed graph, or a directed graph, as appropriate. Suppose the following graph is a subgraph of the Facebook graph.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/20%3A_Counting/20.07%3A_ExercisesRecall that if \(A\) is a finite set with \(\vert A \vert = n\text{,}\) then \(\vert \mathscr{P}(A) \vert = 2^n\text{.}\) Use the Multiplication Rule to verify this formula by considering the construc...Recall that if \(A\) is a finite set with \(\vert A \vert = n\text{,}\) then \(\vert \mathscr{P}(A) \vert = 2^n\text{.}\) Use the Multiplication Rule to verify this formula by considering the construction of an arbitrary subset of \(A\) as a process of making \(n\) “either-or” decisions. For each allowable word of length \(n - 1\) you can create a word of length \(n\) by adding a new letter onto the end.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/22%3A_Combinations/22.02%3A_BasicsCombination: a finite subset of a given set
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/07%3A_Proof_by_mathematical_induction/7.04%3A_ActivitiesComplete the induction step by assuming that the \(n = k\) version of the statement is true, and using this assumption to prove that the \(n = k + 1\) version of the statement is true. p_{1} \wedge p_...Complete the induction step by assuming that the \(n = k\) version of the statement is true, and using this assumption to prove that the \(n = k + 1\) version of the statement is true. p_{1} \wedge p_{2} \wedge &\cdots \wedge p_{n}\\ \hline \left(q_{1} \rightarrow r_{1}\right) &\wedge\left(q_{2} \rightarrow r_{2}\right) \wedge \cdots \wedge\left(q_{n} \rightarrow r_{n}\right)
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/09%3A_Sets/9.06%3A_Alphabets_and_wordsany set can be considered an alphabet
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/07%3A_Proof_by_mathematical_induction/7.01%3A_Principle_of_Mathematical_InductionNext, assume that \(k\) is a fixed number such that \(k \ge 1\text{,}\) and that the statement obtained from the universally quantified predicate is true for \(n = k\text{.}\) Based on this assumption...Next, assume that \(k\) is a fixed number such that \(k \ge 1\text{,}\) and that the statement obtained from the universally quantified predicate is true for \(n = k\text{.}\) Based on this assumption, try to prove that the next case, \(n=k+1\text{,}\) is also true. \begin{align*} (k+1)^3 + (k+2)^3 + (k+3)^3 & = (k^3 + (k+1)^3 + (k+2)^3) + (k+3)^3 - k^3 \\ & = 9m + (k+3)^3 - k^3 \\ & = 9m + (k^3 + 9k^2 + 27k + 27) - k^3 \\ & = 9(m + k^2 + 3k + 3) \text{.} \end{align*}
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Foundations%3A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)/06%3A_Definitions_and_proof_methods/6.05%3A_Statements_Involving_DisjunctionFirst, let's consider a conditional statement with a disjunction on the hypothesis side.
