Reasoning And Logic at Delft University Of Technology | Flashcards & Summaries

# Lernmaterialien für Reasoning and logic an der Delft University of Technology

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TESTE DEIN WISSEN
When are 2 propositions logically equivalent?
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TESTE DEIN WISSEN
• When they have the same value in the final column of their truth tables
( propositions can be logically equivalent even if they have different atoms or not the same number of atoms )

• Also, P & Q are logically equivalent iff P <-> Q is a tautology
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TESTE DEIN WISSEN
What is the order of operators?
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TESTE DEIN WISSEN
*1st to last to evaluate*
Not
And
Or/xor
Implication
Bi-implication

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TESTE DEIN WISSEN
CONTRAPOSITIVE of p -> q
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TESTE DEIN WISSEN
!q -> !p

(And they are equivalent !!!! )
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TESTE DEIN WISSEN
CONVERSE of p -> q
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TESTE DEIN WISSEN
q -> p
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TESTE DEIN WISSEN
INVERSE of p -> q
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TESTE DEIN WISSEN
!p -> !q

(Esti pe invers! E cv negativ:) deci doar negam)
Lösung ausblenden
TESTE DEIN WISSEN
Modus ponens and Modus tollens form? And how can you tell if an argument is valid from these?
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TESTE DEIN WISSEN
Modus ponens:
p -> q
p
Therefore q

Modus tollens:
p -> q
!q
Therefore !p

An argument is valid if the conjunction of the premises logically implies the conclusion:

• Modus ponens: if (p -> q) AND (p)  )-> q  is a tautology then the argument is valid
• Modus tollens: if (p -> q) AND (!q)  )-> !p is a tautology then the argument is valid
Lösung ausblenden
TESTE DEIN WISSEN
When is an argument valid and when is it invalid?
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TESTE DEIN WISSEN
• Valid: if in all cases where the premises are true, the conclusion si also true
• Invalid: if there is at least one case where all the premises are true but the conclusion is false => give counterexample
Lösung ausblenden
TESTE DEIN WISSEN
What does the principle of explosion state?
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TESTE DEIN WISSEN
If we have a contradiction among the premises then the argument is always valid.
(this is since we will never find a case of all premises true but conclusion false to make the arg invalid)
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TESTE DEIN WISSEN
How can an argument be valid if it has no premises?
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TESTE DEIN WISSEN
If the conclusion is a tautology
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TESTE DEIN WISSEN
CNF & DNF
Lösung anzeigen
TESTE DEIN WISSEN
Form truth table and make K-map
• Group all 1s => DNF
• Group all zeros and then negate everything => CNF

DNF = disjunctions of conjunctions
(p AND q) OR (q AND !r)

CNF =  conjunctions of disjunctions
(p OR q) AND (q OR !r)

Both CNF and DNF:
• p/q/ !r …
• a OR b OR c
• a AND b AND c
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TESTE DEIN WISSEN
Necessary and sufficient
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TESTE DEIN WISSEN
• p is necessary for q:   q -> p holds
• p is sufficient for q:   p -> q holds

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TESTE DEIN WISSEN
Proof by generalisation
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TESTE DEIN WISSEN
Used when we have “for all” statements usually.

“Let k be an arbitrary (…)”
Now we assume the premise and and prove that the conclusion follows from that.

“Since k was arbitrarily chosen, it holds that… “

QED (never forget to end your proof)
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Q:
When are 2 propositions logically equivalent?
A:
• When they have the same value in the final column of their truth tables
( propositions can be logically equivalent even if they have different atoms or not the same number of atoms )

• Also, P & Q are logically equivalent iff P <-> Q is a tautology
Q:
What is the order of operators?
A:
*1st to last to evaluate*
Not
And
Or/xor
Implication
Bi-implication

Q:
CONTRAPOSITIVE of p -> q
A:
!q -> !p

(And they are equivalent !!!! )
Q:
CONVERSE of p -> q
A:
q -> p
Q:
INVERSE of p -> q
A:
!p -> !q

(Esti pe invers! E cv negativ:) deci doar negam)
Q:
Modus ponens and Modus tollens form? And how can you tell if an argument is valid from these?
A:
Modus ponens:
p -> q
p
Therefore q

Modus tollens:
p -> q
!q
Therefore !p

An argument is valid if the conjunction of the premises logically implies the conclusion:

• Modus ponens: if (p -> q) AND (p)  )-> q  is a tautology then the argument is valid
• Modus tollens: if (p -> q) AND (!q)  )-> !p is a tautology then the argument is valid
Q:
When is an argument valid and when is it invalid?
A:
• Valid: if in all cases where the premises are true, the conclusion si also true
• Invalid: if there is at least one case where all the premises are true but the conclusion is false => give counterexample
Q:
What does the principle of explosion state?
A:
If we have a contradiction among the premises then the argument is always valid.
(this is since we will never find a case of all premises true but conclusion false to make the arg invalid)
Q:
How can an argument be valid if it has no premises?
A:
If the conclusion is a tautology
Q:
CNF & DNF
A:
Form truth table and make K-map
• Group all 1s => DNF
• Group all zeros and then negate everything => CNF

DNF = disjunctions of conjunctions
(p AND q) OR (q AND !r)

CNF =  conjunctions of disjunctions
(p OR q) AND (q OR !r)

Both CNF and DNF:
• p/q/ !r …
• a OR b OR c
• a AND b AND c
Q:
Necessary and sufficient
A:
• p is necessary for q:   q -> p holds
• p is sufficient for q:   p -> q holds

Q:
Proof by generalisation
A:
Used when we have “for all” statements usually.

“Let k be an arbitrary (…)”
Now we assume the premise and and prove that the conclusion follows from that.

“Since k was arbitrarily chosen, it holds that… “

QED (never forget to end your proof)

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