logic notes
Post on 20-Jan-2015
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Logic
Statements• Statement – a sentence that is either true or
false• Examples:
– Lansing is the Capitol of Michigan– All swimming pools are rectangles– Mr. Cavis is an amazing teacher– Class will be cancelled next Wednesday– 2 is an even number– 13 is an even number
• We often use ‘P’ or ‘Q’ to represent statements– Ex – P1: Lansing is the Capitol of Michigan
P2: All swimming pools are rectangles
Statements – Simple and Compound
• A Simple Statement is a statement that conveys 1 idea
• A Compound Statement is a statement that combines 2 or more simple statements
• Examples:– Mr. Cavis drives a minivan– Seven times four is 28 and today is Friday– The earth is flat or I had waffles for
breakfast
Truth Values and Open Sentences
• A statement’s Truth Value is whether it is true (T) or false (F)
• So P1: Lansing is the Capitol of Michigan has a truth value of true (T)
• While P2: 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.
Truth Tables• Truth Tables are a way of organizing the possible truth values of a statement or series of statements
P
T
F
Q
T
F
P Q
T T
T F
F T
F F
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
Truth Table for Negation
P
T
F
~P
F
T
“And Statements” (Conjunctions)
• 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’)
• Example:– P: I found $5– Q: I crashed my car into a telephone pole– P^Q:
I found $5 AND I crashed my car into a telephone pole.
Truth Table for “And”
• A conjunction is only true if all of the statements in it are true, otherwise it is false
P Q P^Q
T T T
T F F
F T F
F F F
“Or Statements” (Disjunctions)
• When we are making the disjunction of 2 or more statements, we use the word “Or,” and the symbol that we use is ‘V’
• Example:– P: The number 3 is odd– Q: 57 is a prime number– PVQ:
The number 3 is odd OR 57 is a prime number.
Truth Table for “Or”
• A disjunction is true if at least one of the statements in it are true, otherwise it is false.
P Q PVQ
T T T
T F T
F T T
F F F
Implication
• Called an implication because we are “Implying” something to be true
• Also known as an “If-Then” Statement
• An implication for statements P and Q is denoted P=> Q
• An implication is read either “If P, then Q” or “P implies Q”
Truth Table for “If-Then”
• An implication is only false when the first statement is true and the second one is false, otherwise it is true.
P Q P=>Q
T T T
T F F
F T T
F F T
Example of an “If-Then”-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?”- I tell that student “If you get an ‘A’
on the final exam, then you will get an ‘A’ in the class.”
-So here are our 2 statements*P: You get an ‘A’ on the Final Exam*Q: You get an ‘A’ in the class
Example of an “If-Then” (Cont.)
-Think of the combinations of outcomes as if I was telling the truth to that student or not and then consider the possible outcomes:1)Both P and Q are true
- The student got an ‘A’ on the exam and then received an ‘A’ in the class- Therefore, I was telling the truth about the student’s final grade
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 2nd row of the truth table
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
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.
Converse (Not the shoe brand)
-The converse is when you take an “If-Then” statement (P=>Q) and reverse the order of the statements (Q=>P)
*So, Q=>P is the converse of P=>Q- Example:
*Let this be an implication about a triangle ‘T’:
- If T is equilateral, then T is isosceles*So the converse would be:
- If T is Isosceles, then T is equilateral
- Note that the implication (If-Then) is true in this case, but the converse is not.
Biconditional• A biconditional of statements P and
Q is denoted P<=>Q and is read “P if and only if Q”
• A biconditional is nothing more than an “if-then” statement joined with its converse by an “And” – [(P=>Q)^(Q=>P)]
• Note: the prefix “bi” means 2, so biconditional means “2 conditionals (If-Then)’
Truth Tables for Biconditional
- We will work out the 1st truth table in order to complete the
bottom oneP Q P<=>Q
T T T
T F F
F T F
F F T
- Note: A Biconditional is only true when the truth values of ‘P’ and ‘Q’ are the same
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