basic LOGICAL SYMBOLS IN DISCRETE MATHEMATICS (1).pptx

COSC 50A

1. LOGICAL SYMBOLS IN DISCRETE MATHEMATICS

2. The rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc. 2

3. Propositional Logic 3

4. Propositional Logic 4 is concerned with statements to which the truth values, “true” and “false”, can be assigned. The purpose is to analyze these statements either individually or in a composite manner.

5. Propositional Logic 5 A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc ). The connectives connect the propositional variables. Some examples of Propositions are given below "Man is Mortal", it returns truth value “TRUE” "12 + 9 = 3 – 2", it returns truth value “FALSE” The following is not a Proposition − "A is less than 2". It is because unless we give a specific value of A, we cannot say whether the statement is true or false.

6. Propositional Variables 6 The lower case letters starting from P onwards are used to represent propositions Example: p: India is in Asia q: 2 + 2 = 4

7. Compound Statements 7 Statements or propositional variables can be combined by means of logical connectives (operators) to form a single statement called compound statements. The five logical connectives are: Symbol Connective Name ~ Not Negation ∧ And Conjunction ∨ Or Disjunction ⟶ Implies or if...then Implication or conditional ⟷ If and only if Equivalence or biconditional

8. DISJUNCTION also called OR (∨) 8 The OR operation of two propositions A and B (written as A∨B) is true if at least any of the propositional variable A or B is true. The truth table is as follows A B A ∨ B True True True True False True False True True False False False

9. CONJUNCTION also called AND (∧) 9 The AND operation of two propositions A and B (written as A∧B) is true if both the propositional variable A and B is true. The truth table is as follows A B A ∧ B True True True True False False False True False False False False

10. Negation (¬) (~) 10 The negation of a proposition A (written as ¬A) is false when A is true and is true when A The truth table is as follows is false. A ¬ A True False False True

11. Implication / if-then (→) 11 the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true. The truth table is as follows A B A → B True True True True False False False True True False False True

12. 12 A⇔B is bi-conditional logical connective which is true when p and q are same, i.e. both are false or both are true. The truth table is as follows If and only if (⇔) A B A ⇔ B True True True True False False False True False False False True

13. Activity 1 13 Use basic logical symbol in the problems provided below Problem: Let P be the statement "It is raining" and Q be the statement "The ground is wet.“(10 points )

14. Answer 14 Simple problems: And: Is it raining and the ground is wet? (p ∧ q) Or: Is it raining or the ground is wet? ( p ∨ q) Negation: Is it not raining? (P ¬ q ) ( ~ P ~ q ) Implication: If it is raining, then the ground is wet. (p ⟹q) If and only if:  It is raining if and only if the ground is wet. (p ⇔ q ) If and only if: If It is raining and only if the ground is wet (p ⇔ q)