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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/05%3A_Graph_Theory/5.11%3A_Directed_GraphsThus, the entire sum S has value ∑e∈E+sf(e)−∑e∈E−sf(e). On the other hand, we can write the sum S as \[ \sum_{v\in U}\sum_{e\in E_v^+}f(e)- \sum_{v\in U}\su...Thus, the entire sum S has value ∑e∈E+sf(e)−∑e∈E−sf(e). On the other hand, we can write the sum S as ∑v∈U∑e∈E+vf(e)−∑v∈U∑e∈E−vf(e). Every arc e=(x,y) with both x and y in U appears in both sums, that is, in ∑v∈U∑e∈E+vf(e), when v=x, and in ∑v∈U∑e∈E−vf(e), when v=y, and so the flow in such arcs contributes \(…
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/03%3A_Proof_Techniques_I/3.06%3A_Proofs_and_Disproofs_of_Existential_StatementsFrom a certain point of view, there is no need for the current section. If we are proving an existential statement we are disproving some universal statement. (Which has already been discussed.) Simil...From a certain point of view, there is no need for the current section. If we are proving an existential statement we are disproving some universal statement. (Which has already been discussed.) Similarly, if we are trying to disprove an existential statement, then we are actually proving a related universal statement. Nevertheless, sometimes the way a theorem is stated emphasizes the existence question over the corresponding universal.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_Through_Guided_Discovery_(Bogart)/01%3A_What_is_Combinatorics/1.02%3A_Basic_Counting_PrinciplesIn this section, we explore the basic counting principles through a plethora of examples and exercises. One of our goals in these notes is to show how most counting problems can be recognized as count...In this section, we explore the basic counting principles through a plethora of examples and exercises. One of our goals in these notes is to show how most counting problems can be recognized as counting all or some of the elements of a set of standard mathematical objects. You may have noticed some standard mathematical words and phrases such as set, ordered pair, function, and so on creeping into the problems.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.03%3A_Equivalence_RelationsThe main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For examp...The main idea of an equivalence relation is that it is something like equality, but not quite. Usually there is some property that we can name, so that equivalent things share that property. For example Albert Einstein and Adolf Eichmann were two entirely different human beings, if you consider all the different criteria that one can use to distinguish human beings there is little they have in common.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_in_Society_(Lippman)/07%3A_Scheduling/7.02%3A_Getting_StartedThe number in parentheses after the task name is the time required for the task. For simplicity, we are going to make the very big assumptions that every processor can do every task, that they all wou...The number in parentheses after the task name is the time required for the task. For simplicity, we are going to make the very big assumptions that every processor can do every task, that they all would take the same time to complete it, and that only one processor can work on a task at a time. The critical time is the absolute minimum time to complete the job, regardless of the number of processors working on the tasks.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.01%3A_RelationsA relation in mathematics is a symbol that can be placed between two numbers (or variables) to create a logical statement (or open sentence). The main point here is that the insertion of a relation sy...A relation in mathematics is a symbol that can be placed between two numbers (or variables) to create a logical statement (or open sentence). The main point here is that the insertion of a relation symbol between two numbers creates a statement whose value is either true or false. For example, we have previously seen the divisibility symbol (|) and noted the common error of mistaking it for the division symbol (/).
