Logical Statement.pptx
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This document outlines the topics covered in a first-year mathematics course at Salahaddin University, focusing on logical statements, truth tables, and the rules of logic. Key concepts such as connectives, equivalence, tautology, and quantifiers are defined and explained with examples and rules. The outline serves as a guide for students to understand the foundational principles of mathematical logic.
Logical Statement.pptx
- 1. Salahaddin university /Erbil College of Science Mathematics Department 1𝑠𝑡 Year class
- 2. Outline 1. Def/ Logical statement………………………………... 3 2. Connective statements……………………………….. 4 3. Truth tables………………………………………………. 5 4. Converse,Inverse,Contrapossitive…………………... 6 5. Equivalence……………………………………………... 8 6. Tautology and Contraduction……………………….. 9 7. Algebric Rules for statement forms………………….10 8. Quantifiers………………………………………………. 13 9. Theorm…………………………………………………... 15 9/24/2022 2
- 3. Logical statement Definition: is declarative sentence that is either true or false but not both. REMARK In general Question and comments are not logical statementes. For Example: 1. 3< 5 true statement 2. R> 8 𝑖𝑠 𝑛𝑜𝑡 statement 3. 4+6=12 is false statement 4. 1+1=2 is true statement 5. Today is Friday is false statement 9/24/2022 3
- 4. Example Symbol Means Words ~𝑝 𝑜𝑟 ¬𝑝 ~ 𝑜𝑟 ¬ not Negation and Conjunction or Disjunction If then Conditional If and only if Bi conditional q p q p q p q p 9/24/2022 4
- 5. Truth table Let p and q be two statement then the conjunction, Dis conjunction , conditional and Bi conditional. q p T T T T T T F F F T F T F T F T T F T T F F F F q p q p q p q p 9/24/2022 5
- 6. Consider the proposition 𝑝 → 𝑞 • Its Converse is the proposition 𝑞 → 𝑝 • Its Inverse is the proposition ~𝑝 → ~𝑞 • Its Contrapositive is the proposition ~𝑞 → ~𝑝 ~𝒒 → ~𝒑 ~𝒑 → ~𝒒 𝒒 → 𝒑 𝒑 → 𝒒 ~𝒒 𝒒 ~𝒑 𝒑 T T T T F T F T F T T F T F F T T F F T F T T F T T T T T F T F 9/24/2022 6
- 7. Definition: Two statement p and q are equivalence Which is denoted by p≡q , if they have the same Truth table For example; 𝑝 → 𝑞 ≡ ~𝑝 → 𝑞 and because they have the same truth table q p q p 𝒑 ↔ 𝒒 𝒑 → 𝒒 ~𝒑v𝒒 ~𝒒 𝒒 ~𝒑 𝒑 T T T F T F T F F F T F F T T T T F T T F T T T T F T F 9/24/2022 7
- 8. Definition of tutology: If the statement are always true is said to be tautology. Definition of contraduction: if the statement are always false is said to be contraduction. For example: 𝒑˄~𝒑 𝑝˅~𝑝 ~𝒑 𝒑 F T F T F T T F 9/24/2022 8
- 9. Algebric Rules for statement forms I. Idempotent Rules: II. Commutaitive Rules: III. Associative Rules: p p p p p p p q q p p q q p r q p r q p r q p r q p 9/24/2022 9
- 10. Rules (Continued) IV. Distributive Rules: V. Identity (Bound) Rules: r p q p r q p r p q p r q p T T p or p p T p or p p p F p or p p F F p or p 1 1 1 0 0 0 9/24/2022 10
- 11. Rules (Continued) VI. Complement Rules: VII. DeMorgans Rules: VIII. Double negation: IX. Absorpition Rules: F p p or p p T p p or p p 0 1 q p q p q p q p p p p q p p p q p p 9/24/2022 11
- 12. Prove: q p p q T p q p q p q q p q q q p Left-hand statement Commutative law Distributive law Tautology Identity law Commutative law 9/24/2022 12 q p q q p ) (
- 13. The phrase (for all or for each) is called a universal quantifier and is symbolically written as ∀ . The phrase (for some or there exist) is called an existential quantifier and symbolically written as ∃ . Example: 1. ∀𝑥 ∶ 𝑥 > 0 → 𝑥 − 1 > 0 2. ∃𝑥 ∶ 𝑥 > 0 ˄ 𝑥2 = 81 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 ∃𝑥 > 0 ∶ 𝑥2 = 81 9/24/2022 13
- 14. For example: If 𝐴 = 1,2,3 and 𝑃 𝑥 : 𝑥 < 7 𝑃 1 : 1 < 7 𝑖𝑠 𝑇𝑟𝑢𝑒 𝑃 2 : 2 < 7 𝑖𝑠 𝑇𝑟𝑢𝑒 𝑃 3 : 3 < 7 𝑖𝑠 𝑇𝑟𝑢𝑒 This means that (∀𝑥 ∈ 𝐴, 𝑃(𝑥)) is true Another If 𝐴 = 1,2,4 and 𝑃 𝑥 : 𝑥 < 4 𝑃 1 : 1 < 4 𝑖𝑠 𝑇𝑟𝑢𝑒 𝑃 2 : 2 < 4 𝑖𝑠 𝑇𝑟𝑢𝑒 𝑃 4 : 4 < 4 𝑖𝑠 𝐹𝑎𝑙𝑠𝑒 This means that (∃𝑥 ∈ 𝐴, 𝑃 𝑥 ) 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒 9/24/2022 14
- 15. ) ( ) ( ) ( ) ( . 6 ) ( ) ( ) ( ) ( . 5 ) ( ) ( ) ( ) ( . 4 ) ( ) ( ) ( ) ( . 3 ) ( ) ( ) ( ) ( . 2 ) ( ) ( ) ( ) ( . 1 x q x x p x x q x p x x q x x p x x q x p x x q x x p x x q x p x x q x x p x x q x p x x q x x p x x q x p x x q x x p x x q x p x 9/24/2022 15
- 16. Refrence الرياضيات اسس . كردنى ئاماده : BAHZAD BAHRAM 9/24/2022 16