proposition Logic-1.pptx Discrete Mathematics

Topic from Discrete Mathematics

1. Discrete mathematics topic- Propositional Logic PPT Prepared by Vimal kumar Assistant Professor Mathematics Government Degree College Babrala Gunnaur

2. Proposition ( Statement) A proposition is a declarative sentence that is either true or false , but not both . Example : 1. Delhi is capital of India. Example : 2. 1+3 = 4 Example : 3. 3+3 = 5 Example : 4. Normal glass is unbreakable . Proposition in example 1 and 2 are true. Proposition in example 3 and 4 are false.

3. Examples that are not proposition . 1. Read this carefully 2. these are not declarative sentences . Conventional letters are used for propositional variables. Truth value of a proposition is true and is denoted by T, if it is true proposition. Truth value of a proposition is false and is denoted by F, if it is false proposition. .  

4. Symbols and names used in propositions Symbol Name ~ Negation ˄ Conjunction ˅ Disjunction Conditional Biconditional Exclusive Symbol Name ~ Negation ˄ Conjunction ˅ Disjunction Conditional Biconditional Exclusive

5. Definition 1. Negation Let be a proposition . The negation of is denoted by Example find the negation of the proposition “ today is Monday” and express this in simple english . Answer . “Today is not Monday” or “It is not Monday today”   Truth Table for the Negation of a Proposition Truth Table for the Negation of a Proposition

6. Definition 2. C onjunction Let and be propositions . The conjunction of and denoted by , is the propostion “ and ”. The conjunction is true when both and are true and false otherwise.   Truth table for conjunction of two propositions Truth table for conjunction of two propositions

7. Example 1. Find the conjunction of propositions and where is proposition “today is Monday” and is proposition “ It is a raining day.” Solution: The conjunction of these propositions, is the proposition “Today is Monday and it is raining today” this proposition is true on rainy Monday and is false on any day that is not Monday and on Monday when it is not rain.  

8. Definition 3. Disjunction Let and be propositions . The disjunction of and denoted by , is the propostion “ ”. The disjunction is false when both and are false and true otherwise.   Truth table for disjunction of two propositions Truth table for disjunction of two propositions

9. Definition 4. Exclusive Let and be propositions . The exclusive or of and denoted is the propostion that is true when exactly one of and is true and is false otherwise.   Truth table for the exclusive or of two propositions Truth table for the exclusive or of two propositions

10. Definition 5. Conditional Let and be propositions . The conditional statement is the proposition “ If then the conditional statement is false when p is true and q is false , and true otherwise .In conditional statement is called the hypothesis and q is called the conclusion.   Truth table for the conditional statement

11. Example 2 . Let be the statement “ Agrima learns discrete mathematics” and the statement “Agrima will find a good job.” Express the statement as a statement in English. Solution : from definition of conditional statement “ If Agrima learns discrete mathematics, then she will find a good job”. Or “Agrima will find a good job when she learns discrete mathematics.”  

12. Definition 6. Biconditional Let and be propositions . The biconditional statement is the proposition “ if and only the biconditional statement is true when p and q have the same truth values, and false otherwise. Biconditional statements are also called bi-implications .   Truth table for biconditional statement

13. Example 3. let p be the statement “you can take the flight” and let q be the statement “ you buy a ticket”. Solution. The statement is You can take the flight if and only if you can buy a ticket” . Note the truth values of has the same truth values as   Truth table of and

14. Definition 7. Converse For conditional statement The proposition is converse of .   Truth table for converse of

15. Definition 8. Contrapositive for conditional statement The proposition is contrapositive of . Note : same truth vales for the contrapositive of   Truth table for Contrapositive of

16. Definition 9. Inverse for conditional statement The proposition is inverse of   Truth table for Inverse of

17. Definition 10. Tautology A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called tautology. Truth table of tautology Truth table of tautology

18. Definition 10. Contradiction A compound proposition that is always false, no matter what the truth values of the propositions that occur in it, is called contradiction. Truth table of contradiction Truth table of contradiction

19. Definition 11. Predicate A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables values. Example. denote the statement “ ” what are the truth values of and ? Solution . is “4 ” , which is true and is “2 ” , which is false .  

20. Definition 12. Universal Quantification The universal quantification of is the statement “ for all values of x in the domain”. The notation denotes the universal quantification of . Here is called the universal quantifier .An element for which is false is called a counterexample of  

21. Example1. Let , be the statement “ ” what is the truth value of the quantification , where the domain consists of all real numbers? Solution : Because is true for all real numbers x, the quantification is true. Example 2.Let be the statement “ ” what is the truth value of the quantification , where the domain consists of all real numbers? Solution : is not true for every real number , because for is false i.e. , thus is false  

22. Definition 12. Existential Quantification The existential quantification of is the statement “ ”. The notation denotes the existential quantification of . Here is called the existential quantifier . Example .Let be the statement “ ” what is the truth value of the quantification , where the domain consists of all real numbers? Solution : is false for every real number the existential quantification of which is ,is false