2.2: Logical Equivalence: The Laws of Logic
Example (2.7)
For primitive statement p and q, construct a truth table for each
of the following compound statements.
a) ¬p ∨ q
b) p → q
Here we see that the corresponding truth tables for two
statement ¬p ∨ q and p → q are exactly the same.
Definition (2.2)
Two statement s1 , s2 are said to be logically equivalent, and
we write s1 ⇔ s2 , when the statement s1 is true (respectively,
false) if and only if the statement s2 is true (respectively, false).
From the table, we know that ¬p ∨ q ⇔ p → q.
2.2: Logical Equivalence: The Laws of Logic
Example (2.7)
For primitive statement p and q, construct a truth table for each
of the following compound statements.
a) ¬p ∨ q
b) p → q
Here we see that the corresponding truth tables for two
statement ¬p ∨ q and p → q are exactly the same.
Definition (2.2)
Two statement s1 , s2 are said to be logically equivalent, and
we write s1 ⇔ s2 , when the statement s1 is true (respectively,
false) if and only if the statement s2 is true (respectively, false).
From the table, we know that ¬p ∨ q ⇔ p → q.
2.2: Logical Equivalence: The Laws of Logic
Example (2.7)
For primitive statement p and q, construct a truth table for each
of the following compound statements.
a) ¬p ∨ q
b) p → q
Here we see that the corresponding truth tables for two
statement ¬p ∨ q and p → q are exactly the same.
Definition (2.2)
Two statement s1 , s2 are said to be logically equivalent, and
we write s1 ⇔ s2 , when the statement s1 is true (respectively,
false) if and only if the statement s2 is true (respectively, false).
From the table, we know that ¬p ∨ q ⇔ p → q.
2.2: Logical Equivalence: The Laws of Logic
Example (2.7)
For primitive statement p and q, construct a truth table for each
of the following compound statements.
a) ¬p ∨ q
b) p → q
Here we see that the corresponding truth tables for two
statement ¬p ∨ q and p → q are exactly the same.
Definition (2.2)
Two statement s1 , s2 are said to be logically equivalent, and
we write s1 ⇔ s2 , when the statement s1 is true (respectively,
false) if and only if the statement s2 is true (respectively, false).
From the table, we know that ¬p ∨ q ⇔ p → q.
2.2: Logical Equivalence: The Laws of Logic
Example (2.8)
Construct a truth table for each of the following statements
a) ¬(p ∧ q),
b) ¬p ∨ ¬q,
c) ¬(p ∨ q),
d) ¬p ∧ ¬q,
where p, q are primitive statements.
Here, a crucial difference emerges: The negation of the
conjunction of two primitive statement p, q results in the
disjunction of their negations ¬p, ¬q, whereas the negation of
the disjunction of these same statements p, q is logically
equivalent to the conjunction of their negations ¬p, ¬q.
2.2: Logical Equivalence: The Laws of Logic
Example (2.8)
Construct a truth table for each of the following statements
a) ¬(p ∧ q),
b) ¬p ∨ ¬q,
c) ¬(p ∨ q),
d) ¬p ∧ ¬q,
where p, q are primitive statements.
Here, a crucial difference emerges: The negation of the
conjunction of two primitive statement p, q results in the
disjunction of their negations ¬p, ¬q, whereas the negation of
the disjunction of these same statements p, q is logically
equivalent to the conjunction of their negations ¬p, ¬q.
The Laws of Logic 1/2
For any primitive statements p, q, r , any tautology T0 , and any
contradiction F0 ,
1) ¬¬p ⇔ p
Laws of Double Negation
2)
¬(p ∨ q) ⇔ ¬p ∧ ¬q
¬(p ∧ q) ⇔ ¬p ∨ ¬q
3)
p∨q ⇔q∨p
p∧q ⇔q∧p
4)
p ∨ (q ∨ r ) ⇔ (p ∨ q) ∨ r
p ∧ (q ∧ r ) ⇔ (p ∧ q) ∧ r
Associative Laws
5)
p ∨ (q ∧ r ) ⇔ (p ∨ q) ∧ (p ∨ r )
p ∧ (q ∨ r ) ⇔ (p ∧ q) ∨ (p ∧ r )
Distributive Laws
DeMorgan’s Laws
Commutative Laws
The Laws of Logic 2/2
6)
p∨p ⇔p
p∧p ⇔p
7)
p ∨ F0 ⇔ p
p ∧ T0 ⇔ p
Identity Laws
8)
p ∨ ¬p ⇔ T0
p ∧ ¬p ⇔ F0
Inverse Laws
9)
p ∨ T0 ⇔ T0
p ∧ F0 ⇔ F0
Domination Laws
p ∨ (p ∧ q) ⇔ p
p ∧ (p ∨ q) ⇔ p
Absorption Laws
10)
Remark:
T0 =tautology
F0 =contradiction
Idempotent Laws
2.2: Logical Equivalence: The Laws of Logic
Definition (2.3 Dual)
Let s be a statement. If s contains no logical connectives other
than ∧ and ∨ , then the dual of s, denoted sd , is the statement
obtained from s by replacing each occurrence of ∧ and ∨ by ∨
and ∧, respectively, and each occurrence of T0 and F0 by F0
and T0 , respectively.
Theorem (2.1 The Principle of Duality)
Let s and t be statements that contain no logical connectives
other than ∧ and ∨. If s ⇔ t, then sd ⇔ t d .
2.2: Logical Equivalence: The Laws of Logic
Definition (2.3 Dual)
Let s be a statement. If s contains no logical connectives other
than ∧ and ∨ , then the dual of s, denoted sd , is the statement
obtained from s by replacing each occurrence of ∧ and ∨ by ∨
and ∧, respectively, and each occurrence of T0 and F0 by F0
and T0 , respectively.
Theorem (2.1 The Principle of Duality)
Let s and t be statements that contain no logical connectives
other than ∧ and ∨. If s ⇔ t, then sd ⇔ t d .
2.2: Logical Equivalence: The Laws of Logic
Two substitution rules:
1) Suppose that the compound statement P is a tatulogy. If p
is a primitive statement that appears in P and we replace
each occurrence of p by the same statement q, then the
resulting compound statement P1 is also a tatulogy.
2) Let P be a compound statement where p is an arbitrary
statement that appears in P, and let q be a statement such
that q ⇔ p. Suppose that in P we replace one or more
occurrences of p by q. Then this replacement yields the
compound statement P1 . Under these circumstances
P1 ⇔ P.
2.2: Logical Equivalence: The Laws of Logic
Example (2.12)
Negate and simplify the compound statement (p ∨ q) → r .
Example (2.15)
Verify the following compound statements are logically
equivalent:
a) (p → q) ⇔ (¬q → ¬p)
b) (q → p) ⇔ (¬p → ¬q)
The statement ¬q → ¬p is called the contrapositive of the
implication p → q. The statement q → p is called the converse
of p → q; ¬p → ¬q is called the inverse of p → q.
Simplification of compound statements:
Example (2.16, 2.17, 2.18)
Let p, q, r , t denote primitive statements. Simplify each of the
following compound statements:
a) (p ∨ q) ∧ ¬(¬p ∧ q)
b) ¬[¬[(p ∨ q) ∧ r ] ∨ ¬q]
c) (p ∨ q ∨ r ) ∧ (p ∨ t ∨ ¬q) ∧ (p ∨ ¬t ∨ r )