Dicrete mathematics
MTC-1011 :
Discrete Mathematics I
1 Propositional Logic 1
1.1 Basic Concepts in Logic . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Converse, Contrapositive, and Inverse . . . . . . . . . . . . . 5
1.1.2 Precedence of Logical Operators . . . . . . . . . . . . . . . . . 7
1.2 Propositional Equivalences . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Logical Equivalences . . . . . . . . . . . . . . . . . . . . . . . 10
2 Predicates and Quantifiers 16
2.1 Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 The Universal Quantifier . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 The Existential Quantifier . . . . . . . . . . . . . . . . . . . . 20
2.3 Quantifiers with Restricted Domains . . . . . . . . . . . . . . . . . . 21
2.3.1 Precedence of Quantifiers and Binding Variables . . . . . . . . 21
2.3.2 Logical Equivalences Involving Quantifiers . . . . . . . . . . . 22
2.4 Nested Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 The Order of Quantifiers . . . . . . . . . . . . . . . . . . . . . 27
2.4.2 Translating Mathematical Statements . . . . . . . . . . . . . . 29
2.4.3 Translating from Nested Quantifiers into English . . . . . . . 30
2.4.4 Translating English Sentences into Logical Expressions . . . . 31
2.4.5 Negating Nested Quantifiers . . . . . . . . . . . . . . . . . . . 32
3 Methods of Proofs 37
3.1 Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Rules of Inference for Propositional Logic . . . . . . . . . . . . 39
3.1.2 Rules of Inference for Quantified Statements . . . . . . . . . . 43
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