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Logic Notes

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Logic Notes

1. 1. Logic
2. 2. Statements <ul><li>Statement – a sentence that is either true or false </li></ul><ul><li>Examples: </li></ul><ul><ul><li>Lansing is the Capitol of Michigan </li></ul></ul><ul><ul><li>All swimming pools are rectangles </li></ul></ul><ul><ul><li>Mr. Cavis is an amazing teacher </li></ul></ul><ul><ul><li>Class will be cancelled next Wednesday </li></ul></ul><ul><ul><li>2 is an even number </li></ul></ul><ul><ul><li>13 is an even number </li></ul></ul><ul><li>We often use ‘P’ or ‘Q’ to represent statements </li></ul><ul><ul><li>Ex – P 1 : Lansing is the Capitol of Michigan </li></ul></ul><ul><ul><li> P 2 : All swimming pools are rectangles </li></ul></ul>
3. 3. Statements – Simple and Compound <ul><li>A Simple Statement is a statement that conveys 1 idea </li></ul><ul><li>A Compound Statement is a statement that combines 2 or more simple statements </li></ul><ul><li>Examples: </li></ul><ul><ul><li>Mr. Cavis drives a minivan </li></ul></ul><ul><ul><li>Seven times four is 28 and today is Friday </li></ul></ul><ul><ul><li>The earth is flat or I had waffles for breakfast </li></ul></ul>
4. 4. Truth Values and Open Sentences • A statement’s Truth Value is whether it is true (T) or false (F) • So P 1 : Lansing is the Capitol of Michigan has a truth value of true (T) • While P 2 : All swimming pools are rectangles, has a truth value of false (F) • Open sentence – a sentence whose truth value depends on the value of some variable. • Example: - 3x = 12; is a open math sentence.
5. 5. Truth Tables • Truth Tables are a way of organizing the possible truth values of a statement or series of statements F T P F T Q F F T F F T T T Q P
6. 6. Negation – “Not statements” • Negation – Changing a statement so that it has the opposite meaning and truth values - We generally do this by inserting the word ‘NOT’ - The symbol for negation is ‘~’ and is read “Not” - So if we have a statement P: five plus two is seven; the negation of that would be ~P: five plus two is not seven • Example: P: There is snow on the ground ~P: There is not snow on the ground
7. 7. Truth Table for Negation F T P T F ~P
8. 8. “ And Statements” (Conjunctions) <ul><li>When we are making the conjunction of 2 or more statements, we use the word “And,” and the symbol that we use is ‘^’ (Looks like an A without the middle line – ‘And’ starts with ‘A’) </li></ul><ul><li>Example: </li></ul><ul><ul><li>P: I found \$5 </li></ul></ul><ul><ul><li>Q: I crashed my car into a telephone pole </li></ul></ul><ul><ul><li>P^Q: </li></ul></ul>I found \$5 AND I crashed my car into a telephone pole .
9. 9. Truth Table for “And” <ul><li>A conjunction is only true if all of the statements in it are true , otherwise it is false </li></ul>F F F F T F F F T T T T P^Q Q P
10. 10. “ Or Statements” (Disjunctions) <ul><li>When we are making the disjunction of 2 or more statements, we use the word “Or,” and the symbol that we use is ‘ V ’ </li></ul><ul><li>Example: </li></ul><ul><ul><li>P: The number 3 is odd </li></ul></ul><ul><ul><li>Q: 57 is a prime number </li></ul></ul><ul><ul><li>P V Q: </li></ul></ul>The number 3 is odd OR 57 is a prime number .
11. 11. Truth Table for “Or” <ul><li>A disjunction is true if at least one of the statements in it are true , otherwise it is false. </li></ul>F F F T T F T F T T T T P V Q Q P
12. 12. Implication <ul><li>Called an implication because we are “Implying” something to be true </li></ul><ul><li>Also known as an “If-Then” Statement </li></ul><ul><li>An implication for statements P and Q is denoted P=> Q </li></ul><ul><li>An implication is read either “If P, then Q” or “P implies Q” </li></ul>
13. 13. Truth Table for “If-Then” <ul><li>An implication is only false when the first statement is true and the second one is false , otherwise it is true. </li></ul>T F F T T F F F T T T T P => Q Q P
14. 14. Example of an “If-Then” <ul><li>Suppose a student in here is getting a B+ and asks me “Is there any way for me to get an ‘A’ in this class?” </li></ul><ul><li>I tell that student “If you get an ‘A’ on the final exam, then you will get an ‘A’ in the class.” </li></ul><ul><li>So here are our 2 statements </li></ul><ul><ul><li>*P: You get an ‘A’ on the Final Exam </li></ul></ul><ul><ul><li>*Q: You get an ‘A’ in the class </li></ul></ul>
15. 15. Example of an “If-Then” (Cont.) <ul><li>Think of the combinations of outcomes as if I was telling the truth to that student or not and then consider the possible outcomes: </li></ul><ul><li>Both P and Q are true </li></ul><ul><li>- The student got an ‘A’ on the exam and then received an ‘A’ in the class </li></ul><ul><li>- Therefore, I was telling the truth about the student’s final grade </li></ul>
16. 16. Example of an “If-Then” (Cont.) 2) P is true, but Q is false - The student got an ‘A’ on the exam and then did not receive an ‘A’ in the class - Therefore, I was not telling the truth about the student’s final grade - What I said was false, which agrees with the 2 nd row of the truth table
17. 17. Example of an “If-Then” (Cont.) 3) P is false and Q is true - The student did not get an ‘A’ on the exam (say they got a ‘B’) and then received an ‘A’ in the class - I did not lie when I spoke with the student initially, so I was telling the truth
18. 18. Example of an “If-Then” (Cont.) 4) Both P and Q are false - The student did not get an ‘A’ on the exam and did not get an ‘A’ in the class - I only promised an ‘A’ in the class if the student got an ‘A’ on the exam, so again I was telling the truth, which agrees with the last row in the truth table.
19. 19. Converse (Not the shoe brand) <ul><li>The converse is when you take an “If-Then” statement (P=>Q) and reverse the order of the statements (Q=>P) </li></ul><ul><ul><li>*So, Q=>P is the converse of P=>Q </li></ul></ul><ul><li>Example: </li></ul><ul><ul><li>*Let this be an implication about a triangle ‘T’: </li></ul></ul><ul><ul><li>- If T is equilateral, then T is isosceles </li></ul></ul><ul><ul><li>*So the converse would be: </li></ul></ul>- If T is Isosceles, then T is equilateral - Note that the implication (If-Then) is true in this case, but the converse is not.
20. 20. Biconditional <ul><li>A biconditional of statements P and Q is denoted P<=>Q and is read “P if and only if Q” </li></ul><ul><li>A biconditional is nothing more than an “if-then” statement joined with its converse by an “And” – [(P=>Q)^(Q=>P)] </li></ul><ul><li>Note: the prefix “bi” means 2, so biconditional means “2 conditionals (If-Then)’ </li></ul>
21. 21. Truth Tables for Biconditional - We will work out the 1 st truth table in order to complete the bottom one - Note: A Biconditional is only true when the truth values of ‘P’ and ‘Q’ are the same T F F F T F F F T T T T P<=>Q Q P