Propositional Logic 2 Flashcards
Logical equivalence means…
compund propositions have the same truth value.
What are the 4 related implications?
- Direct
- Contrapositive
- Inverse
- Converse
If pvq (direct), what is its contrapositive?
~qv~p.
If pvq (direct), what is its inverse?
~pv~q.
If pvq (direct), what is its converse?
qvp.
What is a contrapositive’s relationship to the direct proposition?
It is equivalent to the direct.
What is an inverse’s relationship to the direct proposition?
It is not equivalent to the direct.
What is a converse’s relationship to the inverse proposition?
It is equivalent to the inverse.
What is De Morgan’s Negation of Conjunction?
~pv~q ≡ ~(p^q).
What is De Morgan’s Negation of Disjunction?
~p^~q ≡ ~(pvq).
What are the 3 basic laws and 3 properties of logical equivalence?
- Identity Law
- Domination Rule
- Idempotent Law
- Double Negation
- Commutative
- Associative
What are the 4 advanced laws of logical equivalence?
- Distributive
- De Morgan’s
- Absorption
- Negation/Complement
Identity Law states that…
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 (^).
Write the Identity Law in symbols.
pvF ≡ p;
p^T ≡ p.
Domination Rule states that…
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 (^).
The combinations in Domination Rule can either be a…
tautology or contradiction.
Write the Domination Rule in symbols.
pvT ≡ T;
p^F ≡ F.
Idempotent Law refers to a property of…
certain operations or functions where applying the operation multiple times doesn’t change the result after the first application.
Write the Idempotent Law in symbols.
pvp ≡ p;
p^p ≡ p.
The property of Double Negation states that…
not not p and p have the same truth value.
Write the property of Double Negation in symbols.
~(~p) ≡ p.
Write the Commutative property in symbols.
pvq ≡ qvp;
p^q ≡ q^p.
Write the Associative property in symbols.
(pvq) v r ≡ p v (qvr);
(p^q) ^ r ≡ p ^ (q^r).
Associativity applies only when…
the connectives involved are exclusively ^ or exclusively v.
Write the Distributive Law in symbols.
p v (q^r) ≡ (pvq) ^ (pvr).
De Morgan’s Law describes how mathematical statements are…
related through their opposites.
De Morgan’s Laws relate conjunctions and disjunctions through…
negation.
Write De Morgan’s Law in symbols.
~(pvq) ≡ ~p^~q;
~(p^q) ≡ ~pv~q.
Absorption Law states that if we combine a statement P by…
∨ with the same statement P and one other statement Q, joined with ∧, then the resultant statement will be the first statement P.
Write the Absorption Law in symbols.
p v (p^q) ≡ p;
p ^ (pvq) ≡ p.
Write the Negation/Complement Law in symbols.
pv~p ≡ T;
p^~p ≡ F.
Negation/Complement laws state that a variable…
AND its negation will produce FALSE, while a variable OR its negation will produce TRUE.
Tips in Proving: Change all _ to _ and _.
implications, or’s, and’s
Tips in Proving: When encountering a negation for a _ proposition, use _ _ _.
compound, De Morgan’s Law
Tips in Proving: Change the operators to the same level of _ to apply laws such as _ , _ ,or _.
precedence, Identity, Idempotence, Complement
Tips in Proving: Always follow the _ of _.
level, precedence
What are the 4 more advanced logical equivalences?
- Material Implication
- Contrapositive/Transposition
- Material Equivalence (biconditionality)
- Others
Material Implication is a valid rule of replacement that allows for a…
conditional statement to be replaced by a disjunction in which the antecedent is negated.
Write Material Implication in symbols.
p→q ≡ ~pvq.
Write Contrapositive/Transposition in symbols.
p→q ≡ ~q→~p.
Conditionals = _ ;
Inverse = _.
Contrapositive;
Converse
Material Equivalence means a biconditional statement can be formed if…
both the direct and converse are both true or false.
Write Material Equivalence in symbols.
p↔q ≡ (p→q) ^ (q→p).
Other logical equivalences involve the operators…
EXOR, and biconditionals.
Write the logical equivalence of EXOR in symbols.
(p⊕q) ≡ (p^~q) v (~p^q).
Write the other logical equivalence of biconditionals in symbols.
p↔q ≡ (~pvq) ^ (~qvp).
A limitation of propositional logic is it does not _.
generalize.
A limitation of propositional logic is it needs to _ each _.
propositionalise, situation
A limitation of propositional logic is it cannot easily _ _ between objects.
express similarities
What is an application of logic in data science in terms of programming languages?
Programming languages uses variables to associate a value that can be used for Boolean operators (e.g., if, else if, while, for, etc. ).
What is the application of logic in data science in terms of learning models?
Making proofs verifies that a method will hold for any particular value or assumption.
