Discrete Structures Predicate Logic 2 Dr Muhammad Humayoun

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Discrete Structures Predicate Logic 2 Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer

Discrete Structures Predicate Logic 2 Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore. mhumayoun@ciitlahore. edu. pk https: //sites. google. com/a/ciitlahore. edu. pk/dstruct/ 1

Negation of Quantifiers • 2

Negation of Quantifiers • 2

Negation of Quantifiers • 3

Negation of Quantifiers • 3

Negation of Quantifiers • 4

Negation of Quantifiers • 4

Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain”

Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • Babies are ignorant. 5

Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain”

Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • Babies are ignorant. (Ambiguous) • All/Some babies are ignorant 6

Exercise • 7

Exercise • 7

Exercise • 8

Exercise • 8

Exercise • 9

Exercise • 9

Exercise • 10

Exercise • 10

Exercise • 11

Exercise • 11

Exercise • 12

Exercise • 12

Exercise • 13

Exercise • 13

Exercise • 14

Exercise • 14

Exercise • 15

Exercise • 15

Exercise • 16

Exercise • 16

Exercise • Useful 17

Exercise • Useful 17

Exercise • 18

Exercise • 18

Exercise • 19

Exercise • 19

Exercise • 20

Exercise • 20

Precedence of Quantifiers • 21

Precedence of Quantifiers • 21

Quantifiers with Restricted Domain • 22

Quantifiers with Restricted Domain • 22

Quantifiers with Restricted Domain • 23

Quantifiers with Restricted Domain • 23

Quantifiers with Restricted Domain • 24

Quantifiers with Restricted Domain • 24

Quantifiers with Restricted Domain • 25

Quantifiers with Restricted Domain • 25

Nested Quantifiers • 26

Nested Quantifiers • 26

Nested Quantifiers • 27

Nested Quantifiers • 27

Meanings of multiple quantifiers • 28

Meanings of multiple quantifiers • 28

Meanings of multiple quantifiers • 29

Meanings of multiple quantifiers • 29

Meanings of multiple quantifiers • 30

Meanings of multiple quantifiers • 30

Meanings of multiple quantifiers • 31

Meanings of multiple quantifiers • 31

Example • 33

Example • 33

Example • 34

Example • 34

Example • 35

Example • 35

Example • 36

Example • 36

From Nested Quantifiers to English • 37

From Nested Quantifiers to English • 37

From Nested Quantifiers to English • 38

From Nested Quantifiers to English • 38

From English to Nested Quantifiers • 39

From English to Nested Quantifiers • 39

From English to Nested Quantifiers • 40

From English to Nested Quantifiers • 40

Bound and free variables l A variable is bound if it is known or

Bound and free variables l A variable is bound if it is known or quantified. Otherwise, it is free. l Examples: l P(x) x is free l P(5) x is bound to 5 l x P(x) x is bound by quantifier Reminder: in a proposition, all variables must be bound. 54

Negating Nested Quantifiers • 55

Negating Nested Quantifiers • 55

Do Exercises 56

Do Exercises 56

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