Lecture 2

Question 1

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

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

A compound proposition that is
neither is called a …………… .

Answer 1

tautology
contradiction
contingency

Question 2

p ≡ q denotes that p and q are ………. .

Answer 2

logically equivalent.

Question 3

two propositions are logically equivalent. if p ↔q is a tautology.

Question 4

Logical Equivalences:-
9) conditional disjunction equivalence:-
¬p ∨ q is equivalent to p → q.

Question 5

Logical Equivalences:-
10) Equivalence of an implication and its contrapositive:
p → q ≡ ¬q → ¬p

Question 6

Logical Equivalences:-
11) The negation of an implication:
¬ (p → q) ≡ p ∧ ¬q

Question 7

Logical Equivalences:-
8) De Morgan’s Law:-
¬ (p ^ q) and ¬p v ¬q are logically equivalent

Question 8

Logical Equivalences:-

1) Commutative properties:
p ∨ q ≡ q ∨ p,
p ∧ q ≡ q ∧ p.

Question 9

Logical Equivalences:-
2) Associative properties:
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)

Question 10

Logical Equivalences:-
3) Distributive laws:
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)

Question 11

Logical Equivalences:-
4) Idempotent laws:
p ∨ p ≡ p
p ∧ p ≡ p

Question 12

Logical Equivalences:-
5)Laws of the excluded middle(inverse law):
p ∨ ¬p ≡ T
p ∧ ¬p ≡ F

Question 13

Logical Equivalences:-
6) Identity laws:
p ∨ F ≡ p
p ∧ T ≡ p

Question 14

Logical Equivalences:-
7) Domination laws:
p ∨ T ≡ T
p ∧ F ≡ F

Question 15

A compound proposition is “satisfiable” if its variables makes a tautology or a contingency or its negation is a tautology i.e.:- there is at least one assignment of truth values for p, q, and r that makes it TRUE.

when the compound proposition is FALSE for all assignments of truth values to its variables, the compound proposition is “unsatisfiable”