Logic

Question 1

What is a proposition?

Answer 1

A proposition is a declarative sentence that is either true or false, but not both.

Question 2

What is a compound proposition?

Answer 2

Many mathematical statements are constructed by combining one or more existing propositions; new propositions, called compound propositions, are formed from existing propositions using logical operators.

Question 3

What will the truth table look like if p and q are in conjunction?

Answer 3

TFFF. The conjunction of p and q, denoted by p ∧ q, is the proposition “p and q”. The conjunction p ∧ q is true when both p and q are true and is false otherwise. The ∧ operator for the two propositions p and q can also be understood as min(p,q) where true is represented with 1 and false is represented with 0.

Question 4

What will the truth table look like if p and q are in disconjunction?

Answer 4

TTTF. The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨ q is false when both p and q are false and is true otherwise. The ∨ operator for the two propositions p and q can also be understood as max(p, q) where true is represented with 1 and false is represented with 0.

Question 5

What will the truth table look like if p and q are in an or-statement (the proposition that is true when exactly one of p and q is true and is false otherwise)?

Answer 5

FTTF. The exclusive or of p and q, denoted by p ⊕ q, is the proposition that is true when exactly one of p and q is true and is false otherwise. The ⊕ operator for the two propositions p and q can also be understood as sum(p, q) == 1 where true is represented with 1 and false is represented with 0

Question 6

What will the truth table look like if p and q are in a conditional statement?

Answer 6

TFTT. The conditional statement p =⇒ q is the proposition “if p, then q.” The conditional statement p =⇒ q is false when p is true and q is false, and true otherwise. In the conditional statement p =⇒ q, p is called the hypothesis (or antecedent or premise), and q is called the conclusion (or consequence). The =⇒ operator for the two propositions p and q can also be understood as max(¬(p),q) where true is represented with 1 and false is represented with 0. The statement p =⇒ q is called a conditional statement because p =⇒ q asserts that q is true on the condition that p holds. A conditional statement is also called an implication. The truth table for the conditional statement p =⇒ q is shown in table above. Note that the statement p =⇒ q is true when both p and q are true and when p is false (no matter what the truth value of q is).

Question 7

What will the truth table look like if p and q are in a biconditional statement?

Answer 7

TFFT. The biconditional statement p ⇐⇒ q is the proposition “p if and only if q.” The biconditional statement p ⇐⇒ q is true when p and q have the same truth values, and is false otherwise. Biconditional statements are also called bi-implications. The ⇐⇒ operator for the two propositions p and q can also be understood as |(p − q)| == 0 where true is represented with 1 and false is represented with 0.

Question 8

What is a tautology?

Answer 8

A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a tautology.

Question 9

What is a compound proposition that is always false called?

Answer 9

A contradiction.

Question 10

What is a contingency?

Answer 10

A compound proposition that is neither a tautology nor a contradiction is called a contingency.

Question 11

When is a compound proposition considered logically equivalent?

Answer 11

The compound propositions p and q are called logically equivalent if p ⇐⇒ q is a tautology. The notation p ≡ q denotes that p and q are logically equivalent.

Question 12

Why are the two logical equivalences known as De Morgan’s laws are particularly important?

Answer 12

They tell us how to negate conjunctions and how to negate disjunctions. In particular, the equivalence ¬(p ∨ q) ≡ ¬p ∧ ¬q tells us that the negation of a disjunction is formed by taking the conjunction of the negations of the component propositions. Similarly, the equivalence ¬(p ∧ q) ≡ ¬p ∨ ¬q tells us that the negation of a conjunction is formed by taking the disjunction of the negations of the component propositions.

Question 13

Use De Morgan’s laws to express the negations of “Miguel has a cellphone and he has a laptop computer”

Answer 13

Let p be “Miguel has a cellphone” and q be “Miguel has a laptop computer.” Then “Miguel has a cellphone and he has a laptop computer” can be represented by p ∧ q. Because of the first of De Morgan’s laws, we can express the negation of our original statement as
“Miguel does not have a cellphone or he does not have a laptop computer.”

Question 14

Use De Morgan’s laws to express the negations of “Heather will go to the concert or Steve will go to the concert.”

Answer 14

Let r be “Heather will go to the concert” and s be “Steve will go to the concert.” Then “Heather will go to the concert or Steve will go to the concert” can be represented by r ∨ s. Because of the second of De Morgans laws, we can express the negation of our original statement as
“Heather will not go to the concert and Steve will not go to the concert.”

Question 15

When is a compound proposition satisfiable?

Answer 15

A compound proposition satisfiable if there is an assignment of truth values to its variables that makes it true.

Question 16

When is a compound proposition unsatisfiable?

Answer 16

when the compound proposition is false for all assignments of truth values to its variables, the compound proposition is unsatisfiable. Note that a compound proposition is unsatisfiable if and only if its negation is true for all assignments of truth values to the variables, i.e., if and only if its negation is a tautology.

Question 17

What does quantifiers express?

Answer 17

Quantification expresses the extent, to which a predicate is true over a range of elements. In English, the words all, some, many, none, and few are used in quantifications

Question 18

How is a universal quantification expressed?

Answer 18

he universal quantification of P(x) is the statement

“P(x) for all values of x in the domain.”

Question 19

How is a existential quantification expressed?

Answer 19

“There exists an element x in the domain such that P(x) evaluates to true.”

Question 20

What do the Identity laws say?

Answer 20

p ∧ T ≡ p

p ∨ F≡ p

Question 21

What do the Domination laws say?

Answer 21

p ∨ T ≡ T

p ∧ F ≡ F

Question 22

What do the Idempotent laws say?

Answer 22

p ∨ p ≡ p

p ∧ p ≡ p

Question 23

What do the Double negations laws say?

Answer 23

¬(¬p) ≡ p

Question 24

What do the Commutative laws say?

Answer 24

p ∨ q ≡ q ∨ p

p ∧ q ≡ q ∧ p

Question 25

What do the Associative laws say?

Answer 25

(p∨q)∨r ≡p∨(q∨r)

p∧q)∧r ≡p∧(q∧r

Question 26

What do the Distributive laws say?

Answer 26

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Question 27

What do the De Morgan’s laws say?

Answer 27

¬(p ∧ q) ≡ ¬p ∨ ¬q

¬(p ∨ q) ≡ ¬p ∧ ¬q

Question 28

What do the Absorption laws say?

Answer 28

p ∨ (p ∧ q) ≡ p

p ∧ (p ∨ q) ≡ p

Question 29

What do the Negation laws say?

Answer 29

p ∨ ¬p ≡ T

p ∧ ¬p ≡ F

Question 30

The proposition q =⇒ p is called the ….. of p =⇒ q.

Answer 30

Converse

Question 31

The …… of p =⇒ q is the proposition ¬q =⇒ ¬p.

Answer 31

Contrapositive

Question 32

The proposition ¬p =⇒ ¬q is called the ….. of p =⇒ q.

Answer 32

Inverse

Question 33

A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a …..

Answer 33

Tautology

Question 34

A compound proposition that is neither a tautology nor a contradiction is called a ….

Answer 34

Contingency

Question 35

Explain, without using a truth table, why (p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r) is true when at least one of p, q, and r is true and at least one is false, but is false when all three variables have the same truth value.

Answer 35

The first clause is true if and only if at least one of p, q, and r is true. The second clause is true if and only if at least one of the three variables is false. Therefore the entire statement is true if and only if there is at least one T and one F among the truth values of the variables, in other words, that they don’t all have the same truth value.