Foundations Of Logic

Question 1

p ∧ q

Answer 1

CONJUNCTION “AND” The conjunction p ∧ q is true when both p and q are true and is false otherwise. (Swift: &&)

Question 2

p ∨ q

Answer 2

DISJUNCTION “OR” 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. (Swift: ||)

Question 3

p⊕q

Answer 3

EXCLUSIVE OR “XOR” 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.

Question 4

p → q

Answer 4

CONDITIONAL STATEMENT or IMPLICATION 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).

Question 5

q → p

Answer 5

CONVERSE The proposition q → p is called the converse of p → q. One of the most common logical errors is to assume that the converse or the inverse of a conditional statement is equivalent to this conditional statement.

Question 6

¬q → ¬p

Answer 6

CONTRAPOSITIVE The contrapositive of p → q is the proposition ¬q → ¬p. Only the contrapositive always has the same truth value as p → q.

Question 7

¬p → ¬q

Answer 7

INVERSE The proposition ¬p → ¬q is called the inverse of p → q. One of the most common logical errors is to assume that the converse or the inverse of a conditional statement is equivalent to this conditional statement.

Question 8

p ↔ q

Answer 8

BI-CONDITIONAL or BI-IMPLICATION 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. “p is necessary and sufficient for q” “if p then q, and conversely” “p iff q.” The last way of expressing the biconditional statement p ↔ q uses the abbreviation “iff” for “if and only if.” Note that p ↔ q has exactly the same truth value as (p → q) ∧ (q → p).

Question 9

What is the order of precedence of the following logical operators? → ∨ ↔ ¬ ∧

Answer 9

¬ 1 ∧ 2 ∨ 3 → 4 ↔ 5

Question 10

Fuzzy Logic

Answer 10

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time.

Question 11

Identity Laws

Answer 11

p∧T≡p

p∨F≡p

Question 12

Domination Laws

Answer 12

p∨T≡T

p∧F≡F

Question 13

Idempotent Laws

Answer 13

p∨p≡p

p∧p≡p

Question 14

Double Negation Law

Answer 14

¬(¬p) ≡ p

Question 15

Commutative Laws

Answer 15

p∨q≡q∨p

p∧q≡q∧p

Question 16

Associative Laws

Answer 16

(p∨q)∨r ≡p∨(q∨r)

(p∧q)∧r ≡p∧(q∧r)

Question 17

Distributive Laws

Answer 17

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Question 18

De Morgan’s Laws

Answer 18

¬(p ∧ q) ≡ ¬p ∨ ¬q

¬(p ∨ q) ≡ ¬p ∧ ¬q

When using De Morgan’s laws, remember to change the logical connective after you negate.

Question 19

Absorption Laws

Answer 19

p ∨ (p ∧ q) ≡ p

p ∧ (p ∨ q) ≡ p

Question 20

Negation Laws

Answer 20

p ∨ ¬p ≡ T

p ∧ ¬p ≡ F

Question 21

Tautology

Answer 21

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

Answer 22

A compound proposition that is always false is called a contradiction.