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MC4142 > Propositional Logic 2 > Flashcards

Propositional Logic 2 Flashcards

1
Q

Logical equivalence means…

A

compund propositions have the same truth value.

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2
Q

What are the 4 related implications?

A
  1. Direct
  2. Contrapositive
  3. Inverse
  4. Converse
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3
Q

If pvq (direct), what is its contrapositive?

A

~qv~p.

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4
Q

If pvq (direct), what is its inverse?

A

~pv~q.

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5
Q

If pvq (direct), what is its converse?

A

qvp.

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6
Q

What is a contrapositive’s relationship to the direct proposition?

A

It is equivalent to the direct.

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7
Q

What is an inverse’s relationship to the direct proposition?

A

It is not equivalent to the direct.

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8
Q

What is a converse’s relationship to the inverse proposition?

A

It is equivalent to the inverse.

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9
Q

What is De Morgan’s Negation of Conjunction?

A

~pv~q ≡ ~(p^q).

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10
Q

What is De Morgan’s Negation of Disjunction?

A

~p^~q ≡ ~(pvq).

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11
Q

What are the 3 basic laws and 3 properties of logical equivalence?

A
  1. Identity Law
  2. Domination Rule
  3. Idempotent Law
  4. Double Negation
  5. Commutative
  6. Associative
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12
Q

What are the 4 advanced laws of logical equivalence?

A
  1. Distributive
  2. De Morgan’s
  3. Absorption
  4. Negation/Complement
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13
Q

Identity Law states that…

A

combining a proposition with the logical identity element (T or F) doesn’t change the value of the proposition when F is used with (v) and T for (^).

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14
Q

Write the Identity Law in symbols.

A

pvF ≡ p;
p^T ≡ p.

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15
Q

Domination Rule states that…

A

if you have a proposition and an identity element (T or F) combined with either logical (v) or logical (^), the result is always true for (v) and false for (^).

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16
Q

The combinations in Domination Rule can either be a…

A

tautology or contradiction.

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17
Q

Write the Domination Rule in symbols.

A

pvT ≡ T;
p^F ≡ F.

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18
Q

Idempotent Law refers to a property of…

A

certain operations or functions where applying the operation multiple times doesn’t change the result after the first application.

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19
Q

Write the Idempotent Law in symbols.

A

pvp ≡ p;
p^p ≡ p.

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20
Q

The property of Double Negation states that…

A

not not p and p have the same truth value.

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21
Q

Write the property of Double Negation in symbols.

A

~(~p) ≡ p.

22
Q

Write the Commutative property in symbols.

A

pvq ≡ qvp;
p^q ≡ q^p.

23
Q

Write the Associative property in symbols.

A

(pvq) v r ≡ p v (qvr);
(p^q) ^ r ≡ p ^ (q^r).

24
Q

Associativity applies only when…

A

the connectives involved are exclusively ^ or exclusively v.

25
Q

Write the Distributive Law in symbols.

A

p v (q^r) ≡ (pvq) ^ (pvr).

26
Q

De Morgan’s Law describes how mathematical statements are…

A

related through their opposites.

27
Q

De Morgan’s Laws relate conjunctions and disjunctions through…

A

negation.

28
Q

Write De Morgan’s Law in symbols.

A

~(pvq) ≡ ~p^~q;
~(p^q) ≡ ~pv~q.

29
Q

Absorption Law states that if we combine a statement P by…

A

∨ with the same statement P and one other statement Q, joined with ∧, then the resultant statement will be the first statement P.

30
Q

Write the Absorption Law in symbols.

A

p v (p^q) ≡ p;
p ^ (pvq) ≡ p.

31
Q

Write the Negation/Complement Law in symbols.

A

pv~p ≡ T;
p^~p ≡ F.

32
Q

Negation/Complement laws state that a variable…

A

AND its negation will produce FALSE, while a variable OR its negation will produce TRUE.

33
Q

Tips in Proving: Change all _ to _ and _.

A

implications, or’s, and’s

34
Q

Tips in Proving: When encountering a negation for a _ proposition, use _ _ _.

A

compound, De Morgan’s Law

35
Q

Tips in Proving: Change the operators to the same level of _ to apply laws such as _ , _ ,or _.

A

precedence, Identity, Idempotence, Complement

36
Q

Tips in Proving: Always follow the _ of _.

A

level, precedence

37
Q

What are the 4 more advanced logical equivalences?

A
  1. Material Implication
  2. Contrapositive/Transposition
  3. Material Equivalence (biconditionality)
  4. Others
38
Q

Material Implication is a valid rule of replacement that allows for a…

A

conditional statement to be replaced by a disjunction in which the antecedent is negated.

39
Q

Write Material Implication in symbols.

A

p→q ≡ ~pvq.

40
Q

Write Contrapositive/Transposition in symbols.

A

p→q ≡ ~q→~p.

41
Q

Conditionals = _ ;
Inverse = _.

A

Contrapositive;
Converse

42
Q

Material Equivalence means a biconditional statement can be formed if…

A

both the direct and converse are both true or false.

43
Q

Write Material Equivalence in symbols.

A

p↔q ≡ (p→q) ^ (q→p).

44
Q

Other logical equivalences involve the operators…

A

EXOR, and biconditionals.

45
Q

Write the logical equivalence of EXOR in symbols.

A

(p⊕q) ≡ (p^~q) v (~p^q).

46
Q

Write the other logical equivalence of biconditionals in symbols.

A

p↔q ≡ (~pvq) ^ (~qvp).

47
Q

A limitation of propositional logic is it does not _.

A

generalize.

48
Q

A limitation of propositional logic is it needs to _ each _.

A

propositionalise, situation

49
Q

A limitation of propositional logic is it cannot easily _ _ between objects.

A

express similarities

50
Q

What is an application of logic in data science in terms of programming languages?

A

Programming languages uses variables to associate a value that can be used for Boolean operators (e.g., if, else if, while, for, etc. ).

51
Q

What is the application of logic in data science in terms of learning models?

A

Making proofs verifies that a method will hold for any particular value or assumption.

MC4142 (5 decks)

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