Propositional Logic 2

Question 1

Logical equivalence means…

Answer 1

compund propositions have the same truth value.

Question 2

What are the 4 related implications?

Answer 2

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

Question 3

If pvq (direct), what is its contrapositive?

Answer 3

~qv~p.

Question 4

If pvq (direct), what is its inverse?

Answer 4

~pv~q.

Question 5

If pvq (direct), what is its converse?

Answer 5

qvp.

Question 6

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

Answer 6

It is equivalent to the direct.

Question 7

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

Answer 7

It is not equivalent to the direct.

Question 8

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

Answer 8

It is equivalent to the inverse.

Question 9

What is De Morgan’s Negation of Conjunction?

Answer 9

~pv~q ≡ ~(p^q).

Question 10

What is De Morgan’s Negation of Disjunction?

Answer 10

~p^~q ≡ ~(pvq).

Question 11

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

Answer 11

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

Question 12

What are the 4 advanced laws of logical equivalence?

Answer 12

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

Question 13

Identity Law states that…

Answer 13

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 (^).

Question 14

Write the Identity Law in symbols.

Answer 14

pvF ≡ p;
p^T ≡ p.

Question 15

Domination Rule states that…

Answer 15

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 (^).

Question 16

The combinations in Domination Rule can either be a…

Answer 16

tautology or contradiction.

Question 17

Write the Domination Rule in symbols.

Answer 17

pvT ≡ T;
p^F ≡ F.

Question 18

Idempotent Law refers to a property of…

Answer 18

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

Question 19

Write the Idempotent Law in symbols.

Answer 19

pvp ≡ p;
p^p ≡ p.

Question 20

The property of Double Negation states that…

Answer 20

not not p and p have the same truth value.

Question 21

Write the property of Double Negation in symbols.

Answer 21

~(~p) ≡ p.

Question 22

Write the Commutative property in symbols.

Answer 22

pvq ≡ qvp;
p^q ≡ q^p.

Question 23

Write the Associative property in symbols.

Answer 23

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

Question 24

Associativity applies only when…

Answer 24

the connectives involved are exclusively ^ or exclusively v.

Question 25

Write the Distributive Law in symbols.

Answer 25

p v (q^r) ≡ (pvq) ^ (pvr).

Question 26

De Morgan's Law describes how mathematical statements are...

Answer 26

related through their opposites.

Question 27

De Morgan's Laws relate conjunctions and disjunctions through...

Answer 27

negation.

Question 28

Write De Morgan's Law in symbols.

Answer 28

~(pvq) ≡ ~p^~q; ~(p^q) ≡ ~pv~q.

Question 29

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

Answer 29

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

Question 30

Write the Absorption Law in symbols.

Answer 30

p v (p^q) ≡ p; p ^ (pvq) ≡ p.

Question 31

Write the Negation/Complement Law in symbols.

Answer 31

pv~p ≡ T; p^~p ≡ F.

Question 32

Negation/Complement laws state that a variable...

Answer 32

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

Question 33

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

Answer 33

implications, or's, and's

Question 34

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

Answer 34

compound, De Morgan's Law

Question 35

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

Answer 35

precedence, Identity, Idempotence, Complement

Question 36

Tips in Proving: Always follow the _ of _.

Answer 36

level, precedence

Question 37

What are the 4 more advanced logical equivalences?

Answer 37

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

Question 38

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

Answer 38

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

Question 39

Write Material Implication in symbols.

Answer 39

p→q ≡ ~pvq.

Question 40

Write Contrapositive/Transposition in symbols.

Answer 40

p→q ≡ ~q→~p.

Question 41

Conditionals = _ ; Inverse = _.

Answer 41

Contrapositive; Converse

Question 42

Material Equivalence means a biconditional statement can be formed if...

Answer 42

both the direct and converse are both true or false.

Question 43

Write Material Equivalence in symbols.

Answer 43

p↔q ≡ (p→q) ^ (q→p).

Question 44

Other logical equivalences involve the operators...

Answer 44

EXOR, and biconditionals.

Question 45

Write the logical equivalence of EXOR in symbols.

Answer 45

(p⊕q) ≡ (p^~q) v (~p^q).

Question 46

Write the other logical equivalence of biconditionals in symbols.

Answer 46

p↔q ≡ (~pvq) ^ (~qvp).

Question 47

A limitation of propositional logic is it does not _.

Answer 47

generalize.

Question 48

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

Answer 48

propositionalise, situation

Question 49

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

Answer 49

express similarities

Question 50

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

Answer 50

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

Question 51

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

Answer 51

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