Discrete Mathematics: Logic, Predicates, and Inference, Lecture notes of Discrete Mathematics

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A set is called countable , if it is finite or countably infinite. A set that is not countable is uncountable

A set S is countable if there is a bijection

The sets in the discrete mathematics are often finite or countable, whereas those in continuous mathematics are often uncountable. 8/19/ Indian Institute of Information Technology, Kottayam 2

What is

Discrete Mathematics is the study of discrete objects and the relationships that bind them. Importance of discrete mathematics is that information is stored and manipulated by computing machines in a discrete fashion. Discrete Mathematics provides the mathematical foundation for many computer science courses. Data structures Algorithms Data base Theory Automata Theory^ Compiler Theory Computer Security Operating Systems 8/19/ Indian Institute of Information Technology, Kottayam 4

Discrete Mathematics

Basic Concepts

Digital and Analog Computer

In computing, there are two primary types of computers: digital and analog. Digital computers process information in binary digits or bits, while analog computers process information in continuously varying physical quantities such as voltage or current. Digital computers process the discrete object 0 and 1, whereas analog computers process continuous data(data obtained through measurement). 8/20/2024 Indian Institute of Information Technology, Kottayam 3

What is Discrete Mathematics?

Discrete Mathematics is the study of discrete objects and the relationships that bind them. Importance of discrete mathematics is that information is stored and manipulated by computing machines in a discrete fashion. Discrete Mathematics provides the mathematical foundation for many computer science courses. Data structures Algorithms Data base Theory Automata Theory Compiler Theory Computer Security Operating Systems 8/20/2024 Indian Institute of Information Technology, Kottayam 4

Proposition

A proposition (statement) is a declarative sentence that is either true or false, but not both. Example

1. Delhi is the capital of India [True]
2. 3+5=8 [True]
3. Kottayam is in Karnataka [False] Note The area of logic that deals with propositions is called propositional calculus. Notation P,q,r,s,t are used to denote propositions, T and F are used to denote true and false proposition. 8/20/2024 Indian Institute of Information Technology, Kottayam 6

Atomic Statement Declarative sentences which cannot be further split into simpler sentences are called atomic statements. Example: Rana is a boy Note: Atomic statements do not contain connectives. Five basic connectives 8/20/2024 Indian Institute of Information Technology, Kottayam 7 S.N English Language Logical connectives Type of operator Symbols 1 and Conjuction binary ⋀ 2 or Disjunction binary ⋁ 3 not Negation Unary ¬ 4 If then Implication or conditional binary → 5 If and only if Biconditional binary (^) ⟷

Conjunction(And)

The conjuction of two statement p and q is the statement p⋀q which is read as ‘p and q’. The statement p⋀q has the truth value (T) whenever both p and q have truth value (T) otherwise it has the truth value (F). Truth table Translate the statement ‘Guru and Roy went up the hill’ into symbolic form. Solution: Let p: Guru went up the hill q: Roy went up the hill Then the statement can be written in symbolic form as p⋀q. 8/20/2024 Indian Institute of Information Technology, Kottayam 9 p q p ⋀ q T T T T F F F T F F F F

Disjunction(Or) The disjuction of two statement p and q is the statement p⋁q which is read as ‘p or q’. The statement p⋁q has the truth value (F)only when both p and q have truth value (F) otherwise it has the truth value (T). Truth table Translate the statement ‘There is something wrong with the bulb or with the wiring’ into symbolic form. Solution: Let p: There is something wrong with the bulb q: Wrong with the wiring Then the statement can be written in symbolic form as p⋁q. 8/20/2024 Indian Institute of Information Technology, Kottayam 10 p q p ⋁ q T T T T F T F T T F F F

Negation

The negation of a statement is generally formed by introducing the word ‘not’ at a proper place in the statement. If p denotes a statement, then the negation of p is written as

¬p read as ‘not p’

Truth table p: Mumbai is a city ¬p : Mumbai is not a city 8/20/2024 Indian Institute of Information Technology, Kottayam 12 p ¬p T F F T

Implication(Conditional statements)

Two propositions p and q can be combined to form statements of the form: If p, then q, such a statement is an implication, denoted by p →q. The component p is the hypothesis of the implication and q the conclusion. The statement p →q has a truth value F when q has the truth value F and p has truth value T, otherwise it has truth value T. p q p → q T T T T F F F T T F F T 8/20/2024 Indian Institute of Information Technology, Kottayam 13