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Knowledge
The lecture delves into various proof strategies, including methods to disprove universally quantified statements using counterexamples, and explores proof by cases and the concept of without loss of generality. It also introduces mechanisms for existential proofs, including constructive and non-constructive methods, and emphasizes the importance of proving uniqueness and utilizing backward reasoning in proofs. Overall, it covers a range of strategies that are vital for understanding mathematical proofs.
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Sections
References
ch11.pdfClass Notes
Memorization
What we have learnt
- Counterexamples are essenti...
- Proof by cases allows for t...
- Existential proofs can be c...
Final Test
Revision Tests
What we have learnt
- Counterexamples are essential for disproving universally quantified statements.
- Proof by cases allows for the breakdown of statements into manageable parts to prove universally quantified implications.
- Existential proofs can be conducted through constructive means by providing specific examples or through logical reasoning without concrete examples.
- Uniqueness proofs establish both the existence of a solution and that no other solutions exist.
- Backward reasoning is a strategic approach that works by assuming true conditions to establish conclusions.
Key Concepts
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Term: Counterexample
Definition: An example used to disprove a universally quantified statement by finding an element for which the statement is false.
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Term: Proof by Cases
Definition: A method of proof where a statement is divided into several cases, each of which must be proven true to show the statement as a whole is true.
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Term: Without Loss of Generality (w.l.o.g.)
Definition: A phrase used in proofs indicating that a result can be assumed for one case without affecting the generality of the proof.
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Term: Constructive Proof
Definition: A proof that demonstrates the existence of a mathematical object by providing a specific example.
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Term: Nonconstructive Proof
Definition: A proof that shows the existence of a mathematical object without providing a specific example.
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Term: Uniqueness Proof
Definition: A proof that shows not only the existence of a solution but also that it is the only solution.
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Term: Backward Reasoning
Definition: A proof strategy where one starts by assuming the conclusion to find a true premise that leads back to that conclusion.