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What is a theorem?
It is a statement that can be shown to be true.
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What is a proof?
A proof is an argument that is valid and by its validity it establishes the truth of a theorem.
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That is an Axiom?
- It is things that are true that we don't have to define in a proof such as what and integer is and its properties.
- All other things we use in a prof that are not classified as axioms could be.
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How do we typically prove something?
We take what we are given and from what we are given we can usully work with the related rules or its properties step by step to end up at our conclusion.
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What types of proofs are used in this course?
Direct proof , indirect proofs such as proof by contradiction and proof by induction.
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Give an example with a direct proof using a conditional statement p=>q to prove that p is true. We also know that the statement "p=>q" is true.
- If we know that
- "p=>q" =T
- q=T
- then following the rules of conditionals the only truth value p can hold i T as something false not can lead to something true to make a true statement.
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- Vrf visar (st)^2 detta och inte mn?
- Svar: Man skriver upp det man vat och sedan det man vill visa på. Sedan bevisar man detta genom att ta legit steg från det man vet till det man vill via på.
- Här blir de så att man tar sig till (st)^2
- D¨vet man alltså att nått kan bli sqrd
- Detta nått är dock en integer, detta bevisar pga att två integer som multipliceras blir en integer därför kan man säga att s*t= g och då kan man skriva (g)^2 = mn och då kan vi se hur
- mn är lika med en intiger i sqrd form. i med det kan man kalla mn en perfect square.
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What differ direct proofs from indirect proofs? Examples?
- Direct leads from the premises and by steps of deductions lead to a conclusion.
- Indirect proofs That do not start from the premises and by deduction lead to a conclusion is called indirect proofs and proofs by contradiction is such a proof.
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In a proof by contradiction how would we prove p is true?
- By proving -p is false, and we do that by the following steps:
- Assume -p is true
- Do regular proof steps
- Derive an explicit contradiction such as (r&-r)
- Because of this derivation we can conclude that something true can not lead to something false and therefor the -p must be false, leading to p being true.
- https://www.youtube.com/watch?v=FA6vELkx0Io&list=RDFA6vELkx0Io&start_radio=1
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What is mathematical induction?
- Mathematical induction is a mathematical proof technique.
- It is essentially used to prove that a property P holds for every natural number n,
- i.e. for n = 0, 1, 2, 3, and so on.
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What is a propositional function?
- A propositional function is a sentence expressed in a way that
- would assume the value of true or false, except that within the sentence
- there is a variable (x) that is not defined or specified. This leaves the statement undetermined.
- The sentence may contain several such
- variables
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What are the 5 steps of a induction proof?
- 1. Prove that the statement is true for p(1)
- 2. Assume it is true for k, set n=k
- 3. Based on assumption replace k with k+1
- 4. Now we replace the series with the formula from step 3 to get two formula expressions on both side of = sign.
- 5. Now solve and simplify to show they are equal.
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