𝐺𝑖𝑣𝑒𝑛:
P Q R ~Q (P ~Q) ~R (Q ^ ~R) ~P (~P V ~R)
T F T
F F T
T T F](https://image.slidesharecdn.com/introduction-to-propositional-logic-1-250725113120-66130124/85/Introduction-to-Propositional-Logic-1-pptx-10-320.jpg)
Introduction-to-Propositional-Logic-1.pptx
- 1. Introduction to Propositional Logic Prepared by: Mr. Ian Anthony M. Torrente, LPT
- 2. What is Propositional Logic? • A proposition(statement) is declarative sentence in which is either true or false. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒: 10+15=25 18>5 𝑀𝑎𝑘𝑎𝑡𝑖𝑖𝑠 h 𝑡 𝑒𝑃𝑎𝑟𝑡 𝑜𝑓 𝑉𝑖𝑠𝑎𝑦𝑎𝑠. 𝐹 𝑖𝑛 𝐴𝐹𝑃𝑚𝑒𝑎𝑛𝑠F𝑜𝑙𝑙𝑜𝑤 . 𝑃𝑙𝑒𝑎𝑠𝑒,𝑠𝑡𝑎𝑛𝑑. h 𝑊 𝑒𝑟𝑒𝑑𝑜 𝑦𝑜𝑢𝑙𝑖𝑣𝑒? 𝑇𝑟𝑢𝑒 𝑇𝑟𝑢𝑒 𝐹𝑎𝑙𝑠𝑒 𝐹𝑎𝑙𝑠𝑒 𝑁𝑜𝑡 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 . 𝑁𝑜𝑡𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡.
- 3. 5 Basic Logic Connectives • 1. Negation (~) • If p is a proposition, then negation of p is a proposition which is- 𝑇𝑟𝑢𝑒 h 𝑤 𝑒𝑛𝑝 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒. 𝐹𝑎𝑙𝑠𝑒 h 𝑤 𝑒𝑛𝑝 𝑖𝑠𝑡𝑟𝑢𝑒. h 𝑇𝑟𝑢𝑡 𝑇𝑎𝑏𝑙𝑒 P ~P F T T F F T 𝑆𝑒𝑛𝑡𝑒𝑛𝑐𝑒: 𝐼𝑓 h 𝑡 𝑒𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 𝑝𝑖𝑠:𝐼𝑡 𝑠 𝑐𝑙𝑜𝑢𝑑𝑦 𝑜𝑢𝑡𝑠𝑖𝑑𝑒. h 𝑇 𝑒𝑛 h 𝑡 𝑒𝑛𝑒𝑔𝑎𝑡𝑖𝑜𝑛𝑖𝑠: 𝐼𝑡 𝑠 𝑁𝑂𝑇 𝑐𝑙𝑜𝑢𝑑𝑦 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 .
- 4. • 2. Conjunction ( ^ ) 𝐼𝑓 𝑝 𝑎𝑛𝑑𝑞𝑎𝑟𝑒𝑡𝑤𝑜𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑠 , h 𝑡 𝑒𝑛𝑐𝑜𝑛𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑜𝑓 𝑝 𝑎𝑛𝑑𝑞𝑖𝑠𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 h h 𝑤 𝑖𝑐 𝑖𝑠 −𝑇𝑟𝑢𝑒 h 𝑤 𝑒𝑛 h 𝑏𝑜𝑡 𝑝𝑎𝑛𝑑𝑞 𝑎𝑟𝑒𝑡𝑟𝑢𝑒 h 𝑇𝑟𝑢𝑡 𝑇𝑎𝑏𝑙𝑒 P Q P ^ Q T T T T F F F F F F T F 𝑆𝑒𝑛𝑡𝑒𝑛𝑐𝑒 : 𝑝: 3+7=10 𝑞:𝐼𝑡 𝑠 𝑐𝑙𝑜𝑢𝑑𝑦 𝑜𝑢𝑡𝑠𝑖𝑑𝑒. h 𝑇 𝑒𝑛,𝑐𝑜𝑛𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑜𝑓 𝑝 𝑎𝑛𝑑𝑞𝑖𝑠 (p ^ q) = 3 + 7 = 10 and it`s cloudy outside
- 5. 3. Disjunction (V) 𝐼𝑓 𝑝𝑎𝑛𝑑𝑞𝑎𝑟𝑒𝑡𝑤𝑜𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑠, h 𝑡 𝑒𝑛𝑑𝑖𝑠𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑜𝑓 𝑝𝑎𝑛𝑑𝑞𝑖𝑠𝑎𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 h h 𝑤 𝑖𝑐 𝑖𝑠: −𝑇𝑟𝑢𝑒 h 𝑤 𝑒𝑛 h 𝑒𝑖𝑡 𝑒𝑟 𝑜𝑛𝑒𝑜𝑓 𝑝𝑜𝑟 𝑞𝑜𝑟 h 𝑏𝑜𝑡 𝑎𝑟𝑒𝑡𝑟𝑢𝑒 h 𝑇𝑟𝑢𝑡 𝑇𝑎𝑏𝑙𝑒 p q p v q T F T F T T T T T F F F 𝑆𝑒𝑛𝑡𝑒𝑛𝑐𝑒: 𝐼𝑓 𝑝 𝑎𝑛𝑑𝑞𝑎𝑟𝑒𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 h 𝑤 𝑒𝑟𝑒: 𝑝: 3+10=13 𝑞: 𝐼𝑡 𝑖𝑠𝑐𝑙𝑜𝑢𝑑𝑦 𝑜𝑢𝑡𝑠𝑖𝑑𝑒. h 𝑇 𝑒 𝑑𝑖𝑠𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑜𝑓 h 𝑡 𝑖𝑠𝑝𝑎𝑛𝑑𝑞 𝑖𝑠: 𝑝𝑉 𝑞:3+10=13𝑜𝑟 𝐼𝑡 𝑖𝑠𝑐𝑙𝑜𝑢𝑑𝑦𝑜𝑢𝑡𝑠𝑖𝑑𝑒.
- 6. 4. Conditional () 𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑎𝑙𝑜𝑓 h 𝑡 𝑒𝑡𝑦𝑝𝑒 if p then q 𝑖𝑠𝑐𝑎𝑙𝑙𝑒𝑑𝑎𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙𝑜𝑟 𝑖𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛. −𝐼𝑡 𝑖𝑠𝑡𝑟𝑢𝑒 h 𝑤 𝑒𝑛 h 𝑏𝑜𝑡 𝑝 𝑎𝑛𝑑𝑞𝑎𝑟𝑒𝑡𝑟𝑢𝑒. −𝐼𝑡 𝑖𝑠𝑡𝑟𝑢𝑒 h 𝑤 𝑒𝑛𝑝𝑖𝑠 𝑓𝑎𝑙𝑠𝑒. h 𝑇𝑟𝑢𝑡 𝑇𝑎𝑏𝑙𝑒 p q P q T T T T F F F T T T T T 𝑆𝑒𝑛𝑡𝑒𝑛𝑐𝑒: 𝑝: 6+15=21 𝑞:22>20 (𝑝→𝑞): 𝐼𝑓 6+15=21, h 𝑡 𝑒𝑛,22>20.
- 7. 5. Biconditional ( ) 𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑜𝑓 h 𝑡 𝑒𝑡𝑦𝑝𝑒 − − 𝑝 𝑖𝑓 𝑛𝑎𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑞 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑏𝑖 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑜𝑟 𝑏𝑖 𝑖𝑚𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 −𝐼𝑡 𝑖𝑠𝑡𝑟𝑢𝑒 h 𝑤 𝑒𝑛 h 𝑒𝑖𝑡 𝑒𝑟 h 𝑏𝑜𝑡 𝑝 𝑎𝑛𝑑𝑞𝑎𝑟𝑒𝑡𝑟𝑢𝑒. −𝐼𝑡 𝑖𝑠𝑡𝑟𝑢𝑒 h 𝑤 𝑒𝑛 h 𝑏𝑜𝑡 𝑝 𝑎𝑛𝑑𝑞𝑎𝑟𝑒 𝑓𝑎𝑙𝑠𝑒. . p q (p q) T F F T F F F T F F F T 𝑆𝑒𝑛𝑡𝑒𝑛𝑐𝑒: 𝑝: 7+21=28 𝑞:21<28<30 (𝑝<−→ 𝑞) :7+21=28 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 21<28<30.
- 8. Try to answer this: 1. ( 𝑃 𝑉 𝑅) 𝑄 2. ( 𝑝𝑟 )→ ( 𝑞 𝑉 𝑝 ) P Q R F T T F T F T T T F F F T F F T F T
- 9. 3 .( 𝑃 𝑉 𝑅) 𝑄 P Q R ~P ~Q ~R (~P V ~R) F T T F T F T T T F F F T F F T F T
- 10. [ ( 𝑃 → 𝑄) 𝑉 (𝑄 𝑅 )]( 𝑃 𝑉 𝑅) 𝐺𝑖𝑣𝑒𝑛: P Q R ~Q (P ~Q) ~R (Q ^ ~R) ~P (~P V ~R) T F T F F T T T F