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Discrete Mathematics

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

2008

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2.2: The Laws of Logic

1.Truth Table Table 2.6

p q ¬p ¬pq pq

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 pq qp (p→q) ∧ (qp) pq

0 0 1 1 1 1

0 1 1 0 0 0

1 0 0 1 0 0

1 1 1 1 1 1

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2.2: The Laws of Logic

1.Truth Table Table 2.6

p q ¬p ¬pq pq

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 pq qp (p→q) ∧ (qp) pq

0 0 1 1 1 1

0 1 1 0 0 0

1 0 0 1 0 0

1 1 1 1 1 1

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Definition (2.2 Logically Equivalent)

Two statements s1,s2are said to be logically equivalent, and we write s1s2,when the statement s1is true(respectively, false) if and only if the statement s2is true(respectively, false).

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The Laws of Logic

1. ¬¬pp Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. pqqp Commutative Laws

pqqp

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

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

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The Laws of Logic

1. ¬¬pp Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. pqqp Commutative Laws

pqqp

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

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

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The Laws of Logic

1. ¬¬pp Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. pqqp Commutative Laws

pqqp

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

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

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The Laws of Logic

1. ¬¬pp Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. pqqp Commutative Laws

pqqp

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

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

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The Laws of Logic

1. ¬¬pp Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. pqqp Commutative Laws

pqqp

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

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

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The Laws of Logic

1. ¬¬pp Laws of Double Negation

2. ¬(p∨q) ⇔ ¬p∧ ¬q DeMorgan’s Laws

¬(p∧q) ⇔ ¬p∨ ¬q

3. pqqp Commutative Laws

pqqp

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

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

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6. ppp Idempotent Laws ppp

7. pF0p Identity Laws pT0p

8. p∨ ¬pT0 Inverse Laws p∧ ¬pF0

9. pT0T0 Domination Laws pF0F0

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

Remark:

T0=tautology F0=contradiction

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6. ppp Idempotent Laws ppp

7. pF0p Identity Laws pT0p

8. p∨ ¬pT0 Inverse Laws p∧ ¬pF0

9. pT0T0 Domination Laws pF0F0

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

Remark:

T0=tautology F0=contradiction

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6. ppp Idempotent Laws ppp

7. pF0p Identity Laws pT0p

8. p∨ ¬pT0 Inverse Laws p∧ ¬pF0

9. pT0T0 Domination Laws pF0F0

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

Remark:

T0=tautology F0=contradiction

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6. ppp Idempotent Laws ppp

7. pF0p Identity Laws pT0p

8. p∨ ¬pT0 Inverse Laws p∧ ¬pF0

9. pT0T0 Domination Laws pF0F0

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

Remark:

T0=tautology F0=contradiction

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6. ppp Idempotent Laws ppp

7. pF0p Identity Laws pT0p

8. p∨ ¬pT0 Inverse Laws p∧ ¬pF0

9. pT0T0 Domination Laws pF0F0

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

Remark:

T0=tautology F0=contradiction

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Definition (2.3)

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 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 thanand.

If st,then sdtd.

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Definition (2.3)

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 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 thanand.

If st,then sdtd.

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Example 1 :

s−(p∧q) ∨ (rT0), sd−(p∨q) ∧ (rF0).

Example 2 :

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

p∧(q∨r) ⇔ (p∧q) ∨ (pr).

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Example 1 :

s−(p∧q) ∨ (rT0), sd−(p∨q) ∧ (rF0).

Example 2 :

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

p∧(q∨r) ⇔ (p∧q) ∨ (pr).

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Example 1 :

s−(p∧q) ∨ (rT0), sd−(p∨q) ∧ (rF0).

Example 2 :

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

p∧(q∨r) ⇔ (p∧q) ∨ (pr).

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Example 1 :

s−(p∧q) ∨ (rT0), sd−(p∨q) ∧ (rF0).

Example 2 :

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

p∧(q∨r) ⇔ (p∧q) ∨ (pr).

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Example (2.18 A switching network) parallel network:pq

series network:pq

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(p∨qr) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬tr)

p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬tr)]

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) ∧ ((¬tq) ∨r)]

p∨[F0∨(¬(¬t∧q) ∧r)]

p∨[(¬(¬t∧q) ∧r)]

p∨[r ∧ ¬(¬t∧q)]

p∨[r ∧(t∨ ¬q)]

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(p∨qr) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬tr)

p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬tr)]

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) ∧ ((¬tq) ∨r)]

p∨[F0∨(¬(¬t∧q) ∧r)]

p∨[(¬(¬t∧q) ∧r)]

p∨[r ∧ ¬(¬t∧q)]

p∨[r ∧(t∨ ¬q)]

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(p∨qr) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬tr)

p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬tr)]

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) ∧ ((¬tq) ∨r)]

p∨[F0∨(¬(¬t∧q) ∧r)]

p∨[(¬(¬t∧q) ∧r)]

p∨[r ∧ ¬(¬t∧q)]

p∨[r ∧(t∨ ¬q)]

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(p∨qr) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬tr)

p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬tr)]

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) ∧ ((¬tq) ∨r)]

p∨[F0∨(¬(¬t∧q) ∧r)]

p∨[(¬(¬t∧q) ∧r)]

p∨[r ∧ ¬(¬t∧q)]

p∨[r ∧(t∨ ¬q)]

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(p∨qr) ∧ (p∨t∨ ¬q) ∧ (p∨ ¬tr)

p∨[(q∨r) ∧ (t∨ ¬q) ∧ (¬tr)]

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) ∧ ((¬tq) ∨r)]

p∨[F0∨(¬(¬t∧q) ∧r)]

p∨[(¬(¬t∧q) ∧r)]

p∨[r ∧ ¬(¬t∧q)]

p∨[r ∧(t∨ ¬q)]

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Thank you.

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