This document summarizes key concepts in propositional logic. It discusses propositional logic, valid arguments, propositions, compound statements formed using logical connectives like conjunction and disjunction, truth tables, formal propositions using negation, conjunction, disjunction, examples of determining if statements are true or false, and conditional statements using implication. It provides examples and truth tables to illustrate these logical concepts and how propositional logic can be used to distinguish valid from invalid arguments.

1. DISCRETE MATHEMATICS
NAME-SURAJ KUMAR
REGD. NO.-210301160019
BY-SASI BHUSAN PADHI
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2. TOPIC--
•Building of valid argument in propositional
logic

3. PROPOSITIONAL LOGIC---
• 1.1- introduction
• 1.2- proposition
• 1.3- compound statement
• 1.4- formal proposition
• 1.5- conditional statement
• 1.6 -Propositional equivalences

4. INTRODUCTION-
• Logic- used to distinguish between valid and invalid
mathematical arguments.
• Logic was developed by Aristotle
• Application in computer science – design computer circuit
,construction of computer program , verification of the
correctness of programs.
• Logic is system based on proposition

5. VALID ARGUMENT
• Defination of valid argument :
• - an argument of valid if whenever the hypotheses are all true , the
conclusion must also be true .
• - that is , when (P1^ P2 ^ ….^ Pn ) Q is a tautology .
• - the previous example had a wff representation of A^b – C WHICH
IS NOT A TAUTOLOGY
• EXAMPLES:-
• - IF GEORGE BUSH IS THE CURRENT PRESIDENT OF THE US , THEN
DICK CHENEY IS THE CURRENT VICE PRESENT

6. PROPOSITION-
• Proposition- is a declarative sentence either true or false, but not
both.
• EG-
• 1) tunku abdul rahman was the first prime prime minister of malaysia
–TRUE
• 2) 1+1=2 – TRUE
• 3) what time is it ? NOT PROPOSITION
• 4) read this carefully . NOT PROPOSITION
• 5) X+1=2 – NOT PROPOSITION
• LETTERS are used to denote propositions- p,q,r,s

7. COMPOUND STATEMENTS
• Many mathematical statement are constructed by
combining one or more propositions.
• EG- JOHN IS SMART OR HE STUDIES EVERY NIGHT
• One or more propositions can be combining to form a
single COMPOUND PROPOSITION using connectives (
logical operator)

8. LOGICAL CONNECTIVES
• CONNECTIVES SYMBOL NAME
• And ^ conjunctions
• Or v disjunction
• Not ~ negation

9. TRUTH TABLE
• Can be used to show how logical operators can
be combine propositions to compound
propositions.
• Displays the truth values that correspond to all
possible value (2^n) of truth values for its
component statements variables.
• The truth value for proposition could be true (T)
OR false (F)

10. FORMAL PROPOSITION
• 1) not ( negation) :~/
• LET p be a proposition. The negation of p is denoted by p, and
read as '' not p''
• - EG: find the negation of proposition ''today is friday'' .
• THE TRUTH TABLE FOR THE NAGATION OF A PROPOSITION
• P _ P
• T F
• F T

11. 2) AND (CONJUNCTION) : ^
• Let p and q be propositions. The proposition of ''p and q'' -
denoted p^q , is TRUE when BOTH p and q are true and other
wise false.
• EG-: student who have taken calculus and computer science
can take this class .
• THE TRUTH TABLE FOR THE CONJUNCTION OF TWO
PROPOSITION:
• P q p^q
• T T T
• T F F
• F T F
• F F F

12. 3) OR (DISJUNCTION): V
• Let p and q be proposition . The preposition of ''p or q'' -
denoted pvq , is FALSE when BOTH p and q are FALSE and TRUE
other wise.
• EG:- student who have taken calculus or computer science can
take this class.
• THE TRUTH TABLE FOR THE DISJUNCTION OF TWO PREPOSITION
• P q pvq
• T T T
• T F T
• F T T
• F F F

13. EXAMPLE 1.1
• Consider the following statement , and determine
whether it is true or false.
• 1) ice floats
• 2) china is in europe and 2+2= 4
• 3) 5-3= 1 or 2 x 2= 4

14. EX-2:
• Let p and q be the following propositions:
• P= it is below freezing
• Q= it is snowing
• Translate the following into logical notation , using p and q and logical
connectives.
• (a) it is below freezing and snowing
• (b) it is below freezing bit not snowing
• (c ) it is not below freezing and it is not snowing
• (d) it is either snowing or below freezing ( or both)

15. CONDITIONAL STATEMENTS
• 1) IMPLICATION
• Let p and q be a proposition . The implication p q is the
proposition that is false when p is true ,q is true , q is false.
Otherwise is TRUE
• P= hypothesis/ antecedent / premise
• Q= conclusion / consequence
• EXPRESS: ''if p , then q'' , ''q when p'',''p implies q''
• EG:- IF you earn an A in logic then I will give you present

16. THANK YOU