proposition Logic-1.pptx 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. 𝑎 + 𝑏 = 𝑐 these are not declarative sentences . Conventional letters 𝑝, 𝑞, 𝑟, 𝑠 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 ꚛ Exclusive
- 5. Definition 1. Negation Let 𝑝 be a proposition . The negation of 𝑝 is denoted by ~𝑝 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 𝑝 ~𝑝 T F F T
- 6. Definition 2. Conjunction Let 𝑝 and 𝑞 be propositions . The conjunction of 𝑝 and 𝑞 denoted by 𝑝˄𝑞 , is the propostion “𝑝 and 𝑞”. The conjunction 𝑝˄𝑞 is true when both 𝑝 and 𝑞 are true and false otherwise. Truth table for conjunction of two propositions 𝑝 𝑞 𝑝˄𝑞 T T T T F F F T F F F F
- 7. Example 1. Find the conjunction of propositions 𝑝 and 𝑞 where 𝑝 is proposition “today is Monday” and 𝑞 is proposition “ It is a raining day.” Solution: The conjunction of these propositions, 𝑝˄𝑞 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 𝑝 and 𝑞 be propositions . The disjunction of 𝑝 and 𝑞 denoted by 𝑝˅𝑞, is the propostion “𝑝 or 𝑞”. The disjunction 𝑝˅𝑞 is false when both 𝑝 and 𝑞 are false and true otherwise. Truth table for disjunction of two propositions 𝑝 𝑞 𝑝˅𝑞 T T T T F T F T T F F F
- 9. Definition 4. Exclusive Let 𝑝 and 𝑞 be propositions . The exclusive or of 𝑝 and 𝑞 denoted 𝑏𝑦 𝑝ꚛ𝑞 is the propostion that is true when exactly one of 𝑝 and 𝑞 is true and is false otherwise. Truth table for the exclusive or of two propositions 𝑝 𝑞 𝑝ꚛ𝑞 T T F T F T F T T F F F
- 10. Definition 5. Conditional Let 𝑝 and 𝑞 be propositions . The conditional statement 𝑝 → 𝑞 is the proposition “ If 𝑝 then 𝑞" the conditional statement 𝑝 → 𝑞 is false when p is true and q is false , and true otherwise .In conditional statement 𝑝 → 𝑞, 𝑝 is called the hypothesis and q is called the conclusion. Truth table for the conditional statement 𝒑 → 𝒒 𝑝 𝑞 𝑝 → 𝑞 T T T T F F F T T F F T
- 11. Example 2 . Let 𝑝 be the statement “ Agrima learns discrete mathematics” and 𝑞 the statement “Agrima will find a good job.” Express the statement 𝑝 → 𝑞 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 𝑝 and 𝑞 be propositions . The biconditional statement 𝑝 ↔ 𝑞 is the proposition “ 𝑝 if and only 𝑞" the biconditional statement 𝑝 ↔ 𝑞 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 𝒑 ↔ 𝒒 𝑝 𝑞 𝑝 ↔ 𝑞 T T T T F F F T F F F T
- 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 𝑝 ↔ 𝑞 statement is You can take the flight if and only if you can buy a ticket” . Note the truth values of 𝑝 ↔ 𝑞 has the same truth values as (𝑝 → 𝑞) ˄ (𝑞 → 𝑝) Truth table of 𝒑 ↔ 𝒒 and (𝒑 → 𝒒) ˄ (𝒒 → 𝒑) 𝑝 𝑞 𝑝 ↔ 𝑞 𝑝 → 𝑞 𝑞 → 𝑝 (𝑝 → 𝑞) ˄ (𝑞 → 𝑝) T T T T T T T F F F T F F T F T F F F F T T T T
- 14. Definition 7. Converse For conditional statement 𝑝 → 𝑞 The proposition 𝑞 → 𝑝 is converse of 𝑝 → 𝑞 . Truth table for converse of 𝒑 → 𝒒 𝑝 𝑞 𝑝 → 𝑞 𝑞 → 𝑝 T T T T T F F T F T T F F F T T
- 15. Definition 8. Contrapositive for conditional statement 𝑝 → 𝑞 The proposition ~𝑞 → ~𝑝 is contrapositive of 𝑝 → 𝑞. Note : same truth vales for the contrapositive of 𝑝 → 𝑞 Truth table for Contrapositive of 𝒑 → 𝒒 𝑝 𝑞 ~𝑞 ~𝑝 𝑝 → 𝑞 ~𝑞 → ~𝑝 T T F F T T T F T F F F F T F T T T F F T T T T
- 16. Definition 9. Inverse for conditional statement 𝑝 → 𝑞 The proposition ~𝑝 → ~𝑞 is inverse of 𝑝 → 𝑞. Truth table for Inverse of 𝒑 → 𝒒 𝑝 𝑞 ~𝑝 ~𝑞 𝑝 → 𝑞 ~𝑝 → ~𝑞 T T F F T T T F F T F T F T T F T F F F T T T T
- 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 𝑃 ~𝑃 𝑃 ˅ ~𝑃 T F T F T T
- 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 𝑃 ~𝑃 𝑃 ˄~𝑃 T F F F T F
- 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. 𝑙𝑒𝑡 𝑝(𝑥) denote the statement “𝑥 > 3” what are the truth values of 𝑝(4) and 𝑝(2) ? Solution . By substituting x = 4,2 𝑝(4) is “4> 3” , which is true and 𝑝(2) is “2> 3” , which is false
- 20. Definition 12. Universal Quantification The universal quantification of 𝑃(𝑥) is the statement “ 𝑝(𝑥) for all values of x in the domain”. The notation ∀𝑥 𝑝(𝑥) denotes the universal quantification of 𝑝(𝑥) . Here ∀ is called the universal quantifier .An element for which 𝑝(𝑥) is false is called a counterexample of ∀𝑥 𝑝(𝑥)
- 21. Example1. Let 𝑝(𝑥), be the statement “ 𝑥 + 1 > 𝑥. ” what is the truth value of the quantification ∀𝑥 𝑝(𝑥), where the domain consists of all real numbers? Solution : Because 𝑝 𝑥 is true for all real numbers x, the quantification ∀𝑥 𝑝 𝑥 is true. Example 2.Let 𝑄 𝑥 be the statement “𝑥 < 2.” what is the truth value of the quantification ∀𝑥𝑄 𝑥 , where the domain consists of all real numbers? Solution : 𝑄 𝑥 is not true for every real number 𝑥, because for 𝑄 3 is false i.e. 3 ≮ 2, x = 3 𝑖𝑠 𝑐𝑜𝑢𝑛𝑡𝑒𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 thus ∀𝑥𝑄 𝑥 is false
- 22. Definition 12. Existential Quantification The existential quantification of 𝑃(𝑥) is the statement “ There exists 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑝(𝑥)”. The notation ∃𝑥 𝑝(𝑥) denotes the existential quantification of 𝑝(𝑥) . Here ∃ is called the existential quantifier . Example .Let 𝑄 𝑥 be the statement “𝑥 = 𝑥 + 1.” what is the truth value of the quantification ∃ 𝑥𝑄 𝑥 , where the domain consists of all real numbers? Solution : 𝑄 𝑥 is false for every real number 𝑥, the existential quantification of 𝑄 𝑥 which is ∃ 𝑥𝑄 𝑥 ,is false