Logic reasoning and set theory pptxxxxxx

Logucy

1. logic Proposition and Sentential Connectives Truth Value and Table Validating Arguments using Truth Table

2. objectives Define logic; Give example of a propositional statement; Distinguish different sentential connectives; Differentiate and give examples of a simple propositional statement from compound propositional statement; Translate logical statements to logical symbols and vice versa ; Construct a truth table; and Validate the arguments.

3. What is logic? The term "logic" came from the Greek word  logos , which is sometimes translated as "sentence", "discourse", "reason", "rule", and "ratio". Of course, these translations are not enough to help us understand the more specialized meaning of "logic" as it is used today . Logic is the study of the principles of correct reasoning . Logic is not the study of psychology of reasoning. The principles of logic are wider but the main (not the only) thing that we study in logic are  principles governing the validity of arguments  - whether certain conclusions follow from some given assumptions . ex. If x > 10, then x > 5. x > 10. therefore: x > 5.

4. proposition Proposition is a propositional statement or an idea which people can  consider  or discuss to decide whether it is true. (Collins English Dict.) A proposition is the basic building block of logic. It is defined as a declarative sentence that is either True (T;1) or False (F;0), but not both . (Introduction to Propositional Logic) A propositional statement can be simple/atomic or compound (w/ the presence of connectives). Examples: 1. I am a teacher. 2. Kim Taehyung is a good dancer. 3. Daniel is an engineer and professional singer. 4. Mharvey is a well-known celebrity but not sociable. 5. If 3x – 2 = 7, then x + 2 = 6.

5. Lets do this. Tell whether the following is a proposition or not. 1. Ellah is a surgeon. 2. What is your name? 3. The teacher is not around. 4. If 2 + 3 = 6, then 3 + 2 = 9. 5. Coco is not a terrorist. 6. Come over here. 7. Buy me a drink. proposition not a proposition proposition proposition proposition not a proposition not a proposition

6. Propositional variable Propositional variable is any letter of the alphabet (but commonly letters used are p, q, r) which represent a proposition/propositional statement. Its purpose may serve in translating logical statements into logical symbols. Logical Statements  Logical Symbols I am a teacher. p Daniel is an engineer and a professional singer. q and r or q  r Hint: Each of the variable corresponds to a simple (atomic) propositional statement. Daniel is an engineer.  q Daniel is a professional singer.  r and   is a connectives

7. Sentential connectives Negation – “not” in symbols “ ” Double negation – “not not ” in symbols “  ” Conjunction – “and” in symbols “ ” Disjunction – “or” in symbols “v” Conditional – “If…, then…” in symbols “, ” Bi-conditional – “if and only if” in symbols “, ” Exclusive or – in symbols “”

8. Other words or phrase may occur in statements Here's a table of some of them and how they are translated . Phrase Logical translation P , but Q P  Q either P or Q P  Q P or Q, but not both P or Q (P  Q)  (  P   Q) OR (P  Q)   ( P  Q) P if Q Q  P P is necessary for Q Q  P P is sufficient for Q P  Q P only if Q P  Q P whenever Q Q  P P is equivalent to Q P  Q Reminders : In logical translation, grouping symbols like ( ), [ ] and { } are used when necessary.

9. Translating logical statements to logical symbols aND VICE VERSA Given below: P : Taliban takes over Afghan government. Q : US soldiers were pulled out from Afghanistan. R : US-Afghan allies are in danger in the hands of the Taliban. S : ISIS grows in numbers. Translate the following: US soldiers were pulled out from Afghanistan but Taliban takes over Afghan government. Q  P

10. Translating logical statements to logical symbols aND VICE VERSA P : Taliban takes over Afghan government. Q : US soldiers were pulled out from Afghanistan. R : US-Afghan allies are in danger in the hands of the Taliban. S : ISIS grows in numbers. If the US soldiers did not pulled out from Afghanistan then ISIS will not grow in numbers. It’s not gonna happen that if US soldiers were pulled out from Afghanistan then US-Afghan allies are in danger in the hands of the Taliban. .  Q   S  (Q  R)

11. logical statements to logical symbols P : Taliban takes over Afghan government. Q : US soldiers were pulled out from Afghanistan. R : US-Afghan allies are in danger in the hands of the Taliban. S : ISIS grows in numbers . ISIS grows in numbers or it’s going to happen that US soldiers were pulled out from Afghanistan is necessary for Taliban will take over Afghan government S  (P  Q)

12. Translating logical symbol to logical statements P : Taliban takes over Afghan government. Q : US soldiers were pulled out from Afghanistan. R : US-Afghan allies are in danger in the hands of the Taliban. S : ISIS grows in numbers.  (S  P) It’s not gonna happen that ISIS grows in numbers if and only if Taliban take over Afghan government. Q  P US soldiers did not pulled out from Afghanistan or Taliban did not take over Afghan government.

13. logical symbols to logical statements P : Taliban takes over Afghan government. Q : US soldiers were pulled out from Afghanistan. R : US-Afghan allies are in danger in the hands of the Taliban. S : ISIS grows in numbers. (S  R)   Q Either if ISIS grows in number then US-Afghan allies are in danger in the hands of the Taliban or US soldiers did not pulled out from Afghanistan. (Q  P)  (S  R) Either if US soldiers pulled out from Afghanistan then Taliban takes over Afghan government or it is possible that if ISIS grows in numbers then US-Afghan allies are in danger in the hands of the Taliban.

14. Truth Value Truth Value is the trut h (T or 1) and falsity (F or 0) of the proposition. The truth value of single/atomic proposition depends on its truthfulness based on facts. ex. Violets are blue. Vinegar is sweet. Viruses are deadly. TRUE TRUE FALSE

15. Truth value of sentential connectives For the truth value of a compound proposition, we need to study the different logical connectives because the truth-value of a compound proposition is a function of, or a quantity dependent upon the truth-values of its component parts . Here are the following Truth Value of sentential connectives in truth table: Negation Double Negation p p T F F T P P P T F T F T F

16. Truth table Disjunction In (P  Q), we need at least one true, either P is true, Q is true or both P and Q are true, then (P  Q) is true. Violets are green or sugar is sour. Chocolate is sweet or violets are blue. FALSE TRUE P Q P  Q T T T T F T F T T F F F

17. Truth table Conjunction To have a truth value of true, we need two (2) trues otherwise it is false. Vinegar is sour and sugar is sweet. Vinegar is sour and sugar is bitter. Vinegar is sweet and sugar is sour. TRUE FALSE FALSE P Q P  Q T T T T F F F T F F F F

18. TRUTH TABLE Conditional . If the “then” statement is true, then the conditional statement is true. If it is false, then “if” statement should be false to make it true otherwise it is false. If 3 + 2 = 5, then 11 – 2 = 9. If 3 x 2 = 5, then 1 x 5 = 5. If 3 + 2 = 6, then 2 x 6 = 3. TRUE TRUE TRUE P Q P  Q T T T T F F F T T F F T

19. TRUTH TABLE Bi-conditional statement If both proposition are true or both are false, then the truth value of bi-conditional statement is true otherwise it is false. Asia is a country if and only if 1 + 1 = 11. Vinegar is sour if and only if sugar is sweet. Vinegar is sour if and only if 1 + 1 = 11. true true false P Q P  Q T T T T F F F T F F F T

20. TRUTH TABLE Exclusive or If both are true or false, then it is false otherwise it is true. It is a complete opposite of the bi-conditional statement. Vinegar is sour exclusively or sugar is sweet. false P Q P  Q T T F T F T F T T F F F

21. LET’S DO THIS: Construct a truth table of the following: (P v Q)  P P Q P v Q (P v Q)  P T T T T T F T T F T T F F F F T

22. TRUTH TABLE (P v Q)   Q P Q P v Q Q (P v Q)  Q T T T F F T F T T T F T T F F F F F T F

23. TRUTH TABLE (P  Q)  (Q  P ) (P  Q)  (P  Q ) P Q P Q Q P QP (P Q)(QP) T T T F F F T T F F T F F F F T T F T F T F F T T T T F P Q P  Q (PQ) P  Q (PQ)v(PQ) T T T F F F T F F T T T F T F T T T F F T F F F

24. TRUTH TABLE ( P  Q)  R P Q R P PQ (PQ)R T T T F F T T T F F F F T F T F T T T F F F T T F T T T T T F T F T T T F F T T F T F F F T F F

25. TAUTOLOGIES Tautologies are different assertions in every possible interpretations. Here are the following tautologies: Tautology - an assertion that is true in every possible interpretation. Example : (A  B)  (A  B) A B A  B (A  B) (A B)  (AB) T T T F T T F T F T F T T F T F F F T T ALL TRUE

26. TAUTOLOGIES Contradiction – an assertion that is false in every possible interpretation Example: (P  Q)  (P  Q) p Q P Q (PQ) (P Q)  (PQ) T T T F F T F T F F F T T F F F F F T F ALL FALSE

27. TAUTOLOGIES Contingency – an assertion that is a combination of true and false in every interpretation. Example: (R  S)  R R S R S R (RS)  R T T T F T T F F F F F T T T T F F T T T COMBINATION OF TRUE & FALSE

28. VALIDATING AN ARGUMENTS USING TRUTH TABLE Steps how to validate an argument using truth table: If the arguments presented is in statement form, assign variables to its identified atomic proposition and translate the arguments into symbolic logic. EXAMPLE 1: Validate the argument below. I will take my lunch today in Korea or in Japan. I did not take my lunch in Korea. Therefore : I did my lunch in Japan. Assign variables to its identified atomic propositional statement: P = I will take my lunch in Korea. Q = I will take my lunch in Japan. P  Q P  Q

29. VALIDATING AN ARGUMENTS USING TRUTH TABLE Steps how to validate an argument using truth table: Make a truth table. P v Q P  Q Decide whether it is valid or not. Look for the truth (2 trues ) of the premises, If all 2 trues leads to the truth (T) of the conclusion then it is valid. However if it leads to a false (F) in the conclusion then it is invalid. AND SO THE SAMPLE ARGUMENT IS VALID . P Q P  Q P Q T T F F T T F T F F F T T T T F F F T F 1 ST PREMISE 2 nd PREMISE CONCLUSION The truths of the premises will lead to the truth of the conclusion

30. VALIDATING AN ARGUMENTS USING TRUTH TABLE SAMPLE 2: If it rains then I will get wet. I get wet. Therefore: It rained. Symbolic logic: P  Q Q  P THE ARGUMENT IS INVALID. P = It rains. Q = I get wet. P Q P Q Q P T T T T T T F F F T F T T T F F F T F F Because the truths of the premises leads to a false conclusion

31. VALIDATING AN ARGUMENTS USING TRUTH TABLE EXAMPLE 3: P  Q Q  R  P  R THE ARGUMENT IS VALID. P Q R P Q Q R P R T T T T T T T T F T F F T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T F F F T T T