Discrete Mathematics

Discrete Mathematics

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

2008

2.2: The Laws of Logic

1.Truth Table Table 2.6

p q ¬p ¬p∨q p→q

0 0 1 1 1

0 1 1 1 1

1 0 0 0 0

1 1 0 1 1

Table 2.7

p q p→q q→p (p→q) ∧ (q→p) p↔q

0 0 1 1 1 1

0 1 1 0 0 0

1 0 0 1 0 0

1 1 1 1 1 1

2.2: The Laws of Logic

1.Truth Table Table 2.6

p q ¬p ¬p∨q p→q

0 0 1 1 1

0 1 1 1 1

1 0 0 0 0

1 1 0 1 1

Table 2.7

p q p→q q→p (p→q) ∧ (q→p) p↔q

0 0 1 1 1 1

0 1 1 0 0 0

1 0 0 1 0 0

1 1 1 1 1 1

Definition (2.2 Logically Equivalent)

Two statements s₁,s₂are said to be logically equivalent, and we write s₁⇔s₂,when the statement s₁is true(respectively, false) if and only if the statement s₂is true(respectively, false).

The Laws of Logic

1. ¬¬p⇔p Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. p∨q⇔q∨p Commutative Laws

p∧q⇔q∧p

4. p∨(q∨r) ⇔ (p∨q) ∨r Associative Laws p∧(q∧r) ⇔ (p∧q) ∧r

5. p∨(q∧r) ⇔ (p∨q) ∧ (p∨r) Distributive Laws p∧(q∨r) ⇔ (p∧q) ∨ (p∧r)

The Laws of Logic

1. ¬¬p⇔p Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. p∨q⇔q∨p Commutative Laws

p∧q⇔q∧p

4. p∨(q∨r) ⇔ (p∨q) ∨r Associative Laws p∧(q∧r) ⇔ (p∧q) ∧r

5. p∨(q∧r) ⇔ (p∨q) ∧ (p∨r) Distributive Laws p∧(q∨r) ⇔ (p∧q) ∨ (p∧r)

The Laws of Logic

1. ¬¬p⇔p Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. p∨q⇔q∨p Commutative Laws

p∧q⇔q∧p

4. p∨(q∨r) ⇔ (p∨q) ∨r Associative Laws p∧(q∧r) ⇔ (p∧q) ∧r

5. p∨(q∧r) ⇔ (p∨q) ∧ (p∨r) Distributive Laws p∧(q∨r) ⇔ (p∧q) ∨ (p∧r)

The Laws of Logic

1. ¬¬p⇔p Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. p∨q⇔q∨p Commutative Laws

p∧q⇔q∧p

4. p∨(q∨r) ⇔ (p∨q) ∨r Associative Laws p∧(q∧r) ⇔ (p∧q) ∧r

5. p∨(q∧r) ⇔ (p∨q) ∧ (p∨r) Distributive Laws p∧(q∨r) ⇔ (p∧q) ∨ (p∧r)

The Laws of Logic

1. ¬¬p⇔p Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. p∨q⇔q∨p Commutative Laws

p∧q⇔q∧p

4. p∨(q∨r) ⇔ (p∨q) ∨r Associative Laws p∧(q∧r) ⇔ (p∧q) ∧r

5. p∨(q∧r) ⇔ (p∨q) ∧ (p∨r) Distributive Laws p∧(q∨r) ⇔ (p∧q) ∨ (p∧r)

The Laws of Logic

1. ¬¬p⇔p Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. p∨q⇔q∨p Commutative Laws

p∧q⇔q∧p

4. p∨(q∨r) ⇔ (p∨q) ∨r Associative Laws p∧(q∧r) ⇔ (p∧q) ∧r

5. p∨(q∧r) ⇔ (p∨q) ∧ (p∨r) Distributive Laws p∧(q∨r) ⇔ (p∧q) ∨ (p∧r)

6. p∨p⇔p Idempotent Laws p∧p⇔p

7. p∨F₀⇔p Identity Laws p∧T₀⇔p

8. p∨ ¬p⇔T₀ Inverse Laws p∧ ¬p⇔F0

9. p∨T₀⇔T₀ Domination Laws p∧F₀⇔F₀

10. p∨(p∧q) ⇔p Absorption Laws p∧(p∨q) ⇔p

Remark:

T₀=tautology F₀=contradiction

6. p∨p⇔p Idempotent Laws p∧p⇔p

7. p∨F₀⇔p Identity Laws p∧T₀⇔p

8. p∨ ¬p⇔T₀ Inverse Laws p∧ ¬p⇔F0

9. p∨T₀⇔T₀ Domination Laws p∧F₀⇔F₀

10. p∨(p∧q) ⇔p Absorption Laws p∧(p∨q) ⇔p

Remark:

T₀=tautology F₀=contradiction

6. p∨p⇔p Idempotent Laws p∧p⇔p

7. p∨F₀⇔p Identity Laws p∧T₀⇔p

8. p∨ ¬p⇔T₀ Inverse Laws p∧ ¬p⇔F0

9. p∨T₀⇔T₀ Domination Laws p∧F₀⇔F₀

10. p∨(p∧q) ⇔p Absorption Laws p∧(p∨q) ⇔p

Remark:

T₀=tautology F₀=contradiction

6. p∨p⇔p Idempotent Laws p∧p⇔p

7. p∨F₀⇔p Identity Laws p∧T₀⇔p

8. p∨ ¬p⇔T₀ Inverse Laws p∧ ¬p⇔F0

9. p∨T₀⇔T₀ Domination Laws p∧F₀⇔F₀

10. p∨(p∧q) ⇔p Absorption Laws p∧(p∨q) ⇔p

Remark:

T₀=tautology F₀=contradiction

6. p∨p⇔p Idempotent Laws p∧p⇔p

7. p∨F₀⇔p Identity Laws p∧T₀⇔p

8. p∨ ¬p⇔T₀ Inverse Laws p∧ ¬p⇔F0

9. p∨T₀⇔T₀ Domination Laws p∧F₀⇔F₀

10. p∨(p∧q) ⇔p Absorption Laws p∧(p∨q) ⇔p

Remark:

T₀=tautology F₀=contradiction

Definition (2.3)

Let s be a statement. If s contains no logical connectives other than∧and∨, then the dual of s, denoted s^d , is the statement obtained from s by replacing each occurrence of∧and∨by∨ and∧, respectively, and each occurrence of T0and F0by 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 s^d ⇔t^d.

Definition (2.3)

Let s be a statement. If s contains no logical connectives other than∧and∨, then the dual of s, denoted s^d , is the statement obtained from s by replacing each occurrence of∧and∨by∨ and∧, respectively, and each occurrence of T0and F0by 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 s^d ⇔t^d.

Example 1 :

s−(p∧q) ∨ (r ∧T₀), s^d−(p∨q) ∧ (r ∨F₀).

Example 2 :

p∨(q∧r) ⇔ (p∨q) ∧ (p∨r), By Thm 2.1, we have

p∧(q∨r) ⇔ (p∧q) ∨ (p∧r).

Example 1 :

s−(p∧q) ∨ (r ∧T₀), s^d−(p∨q) ∧ (r ∨F₀).

Example 2 :

p∨(q∧r) ⇔ (p∨q) ∧ (p∨r), By Thm 2.1, we have

p∧(q∨r) ⇔ (p∧q) ∨ (p∧r).

Example 1 :

s−(p∧q) ∨ (r ∧T₀), s^d−(p∨q) ∧ (r ∨F₀).

Example 2 :

p∨(q∧r) ⇔ (p∨q) ∧ (p∨r), By Thm 2.1, we have

p∧(q∨r) ⇔ (p∧q) ∨ (p∧r).

Example 1 :

s−(p∧q) ∨ (r ∧T₀), s^d−(p∨q) ∧ (r ∨F₀).

Example 2 :

p∨(q∧r) ⇔ (p∨q) ∧ (p∨r), By Thm 2.1, we have

p∧(q∨r) ⇔ (p∧q) ∨ (p∧r).

Example (2.18 A switching network) parallel network:p∨q

series network:p∧q

(p∨q∨r) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬t∨r)

⇔ p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬t∨r)]

⇔ p∨[(q∨r) ∧ (¬t∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (¬¬t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ ¬(¬t∧q)]

⇔ p∨[¬(¬t∧q) ∧ ((¬t∧q) ∨r)]

⇔ p∨[F0∨(¬(¬t∧q) ∧r)]

⇔ p∨[(¬(¬t∧q) ∧r)]

⇔ p∨[r ∧ ¬(¬t∧q)]

⇔ p∨[r ∧(t∨ ¬q)]

(p∨q∨r) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬t∨r)

⇔ p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬t∨r)]

⇔ p∨[(q∨r) ∧ (¬t∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (¬¬t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ ¬(¬t∧q)]

⇔ p∨[¬(¬t∧q) ∧ ((¬t∧q) ∨r)]

⇔ p∨[F0∨(¬(¬t∧q) ∧r)]

⇔ p∨[(¬(¬t∧q) ∧r)]

⇔ p∨[r ∧ ¬(¬t∧q)]

⇔ p∨[r ∧(t∨ ¬q)]

(p∨q∨r) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬t∨r)

⇔ p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬t∨r)]

⇔ p∨[(q∨r) ∧ (¬t∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (¬¬t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ ¬(¬t∧q)]

⇔ p∨[¬(¬t∧q) ∧ ((¬t∧q) ∨r)]

⇔ p∨[F0∨(¬(¬t∧q) ∧r)]

⇔ p∨[(¬(¬t∧q) ∧r)]

⇔ p∨[r ∧ ¬(¬t∧q)]

⇔ p∨[r ∧(t∨ ¬q)]

(p∨q∨r) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬t∨r)

⇔ p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬t∨r)]

⇔ p∨[(q∨r) ∧ (¬t∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (¬¬t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ ¬(¬t∧q)]

⇔ p∨[¬(¬t∧q) ∧ ((¬t∧q) ∨r)]

⇔ p∨[F0∨(¬(¬t∧q) ∧r)]

⇔ p∨[(¬(¬t∧q) ∧r)]

⇔ p∨[r ∧ ¬(¬t∧q)]

⇔ p∨[r ∧(t∨ ¬q)]

(p∨q∨r) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬t∨r)

⇔ p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬t∨r)]

⇔ p∨[(q∨r) ∧ (¬t∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ (¬¬t∨ ¬q)]

⇔ p∨[((q∧ ¬t) ∨r) ∧ ¬(¬t∧q)]

⇔ p∨[¬(¬t∧q) ∧ ((¬t∧q) ∨r)]

⇔ p∨[F0∨(¬(¬t∧q) ∧r)]

⇔ p∨[(¬(¬t∧q) ∧r)]

⇔ p∨[r ∧ ¬(¬t∧q)]

⇔ p∨[r ∧(t∨ ¬q)]

Thank you.