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# Discrete Mathematics

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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.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung