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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/01%3A_Introduction_and_NotationWisdom is the quality that keeps you from getting into situations where you need it. -- Doug Larson
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/05%3A_Proof_Techniques_II_-_Induction/5.01%3A_The_Principle_of_Mathematical_InductionThe Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lift...The Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lifting yourself into the air. Despite the indisputable fact that proofs by PMI often feel like magic, we need to convince you of the validity of this proof technique. It is one of the most important tools in your mathematical kit!
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/04%3A_Sets/4.04%3A_Venn_DiagramsHopefully, you’ve seen Venn diagrams before, but possibly you haven’t thought deeply about them. Venn diagrams take advantage of an obvious but important property of closed curves drawn in the plane. ...Hopefully, you’ve seen Venn diagrams before, but possibly you haven’t thought deeply about them. Venn diagrams take advantage of an obvious but important property of closed curves drawn in the plane. They divide the points in the plane into two sets, those that are inside the curve and those that are outside! (Forget for a moment about the points that are on the curve.) This seemingly obvious statement is known as the Jordan curve theorem, and actually requires some details.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/08%3A_Cardinality/8.05%3A_The_Continuum_Hypothesis_and_The_Generalized_Continuum_HypothesisThe word “continuum” in the title of this section is used to indicate sets of points that have a certain continuity property. For example, in a real interval it is possible to move from one point to a...The word “continuum” in the title of this section is used to indicate sets of points that have a certain continuity property. For example, in a real interval it is possible to move from one point to another, in a smooth fashion, without ever leaving the interval. In a range of rational numbers this is not possible, because there are irrational values in between every pair of rationals.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/06%3A_Relations_and_Functions/6.04%3A_Ordering_RelationsThe prototype for ordering relations is ≤. Although a case could be made for using < as the prototypical ordering relation. These two relations differ in one important sense: ≤ is reflexive and ...The prototype for ordering relations is ≤. Although a case could be made for using < as the prototypical ordering relation. These two relations differ in one important sense: ≤ is reflexive and < is irreflexive. Various authors, having made different choices as to which of these is the more prototypical, have defined ordering relations in slightly different ways. The majority view seems to be that an ordering relation is reflexive (which means that ordering relations are modeled after ≤)
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/08%3A_CardinalityThe very existence of flame-throwers proves that some time, somewhere, someone said to themselves, “You know, I want to set those people over there on fire, but I’m just not close enough to get the jo...The very existence of flame-throwers proves that some time, somewhere, someone said to themselves, “You know, I want to set those people over there on fire, but I’m just not close enough to get the job done.” –George Carlin
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/05%3A_Proof_Techniques_II_-_Induction/5.04%3A_The_Strong_Form_of_Mathematical_InductionThe strong form of mathematical induction (a.k.a. the principle of complete induction, PCI; also a.k.a. course-of-values induction) is so-called because the hypotheses one uses are stronger.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/03%3A_Proof_Techniques_I/3.04%3A_DisproofsThe idea of a “disproof” is really just semantics – in order to disprove a statement we need to prove its negation. If we are given a universally quantified statement the first thing to do is try it o...The idea of a “disproof” is really just semantics – in order to disprove a statement we need to prove its negation. If we are given a universally quantified statement the first thing to do is try it out for some random elements of the universe we’re working in. If we happen across a value that satisfies the statement’s hypotheses but doesn’t satisfy the conclusion, we’ve found what is known as a counterexample.
- 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/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/05%3A_Proof_Techniques_II_-_Induction/5.03%3A_Divisibility_Statements_and_Other_Proofs_Using_PMIThere is a very famous result known as Fermat’s Little Theorem. This would probably be abbreviated FLT except for two things. In science fiction FLT means “faster than light travel” and there is anoth...There is a very famous result known as Fermat’s Little Theorem. This would probably be abbreviated FLT except for two things. In science fiction FLT means “faster than light travel” and there is another theorem due to Fermat that goes by the initials FLT: Fermat’s Last Theorem.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/02%3A_Logic_and_QuantifiersIf at first you don’t succeed, try again. Then quit. There’s no use being a damn fool about it. Fields
