lecture05(complete)

lecture05(completeï&frac...

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MA1100 Lecture 5 Logic Converse and contrapositive Biconditionals autologies and contradictions Tautologies and contradictions Logical equivalence 1 Chartrand: section 2.6 – 2.9
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t Announcement ± Lecture quiz starts today ² Sit with your buddy ² You need to key in both your student numbers sing your clicker (instruction later) using your clicker (instruction later) ± Tutorial begins this week ttendance will be taken ² Attendance will be taken ± Homework set 1 due next Tuesday o not mix up tutorial & HW problem sets ² Do not mix up tutorial & HW problem sets ± Virtual tutorial class terested students see me during lecture break Lecture 5 2 ² Interested students see me during lecture break to fix timeslot
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r ( f r l t l t r ) Summary (from last lecture) tatement pen Sentence Statement Open Sentence Has truth value (T/F) Does not have truth value as no (free) variables as one or more variables We can make an open sentence into a statement by: Has no (free) variables Has one or more variables (i) Substituting its variables with appropriate values (ii) Quantifying the variable (next lecture) Example P(n): n is a prime number (i) P(2): 2 is a prime number Lecture 5 3 (ii) For all n, n is a prime number
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r ( f r l t l t r ) Summary (from last lecture) Operator Standard form Symbolic When is it false? form Negation not P ~ P P true Implication Conjunction P and Q If P then Q P Q P Q P true, Q false One of P, Q false Disjunction P or Q P ¤ Q P if and only if Q P Q Both of P, Q false P, Q have opposite truth values Biconditional Other forms of implication ( necessary, sufficient, only if ) Lecture 5 4
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h r ’ th Tr r ? Where’s the Treasure ? Given five true statements 1. If this house is next to a lake, then the treasure is not in the kitchen. 2. If the tree in the front yard is an elm, then the treasure is in the kitchen. 3. This house is next to a lake. 4. The tree in the front yard is an elm or the easure is buried under the flagpole treasure is buried under the flagpole. 5. If the tree in the back yard is an oak, then the treasure is in the garage. Lecture 5 5 au gaag
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p l i t i Implication Example If 2 is on one side of the card, then A is on the other side p P Q Q P If A is on one side of the card, then 2 i on the other side Do they have the same meaning? so eo e sde Lecture 5 6
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r Converse Q P is called the converse of P Q Q An implication does not have the same meaning as its converse . g Truth table of P Q vs Q P PQ P QQ P TT T T F FT F T Lecture 5 7 F F T
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r It is possible that an implication and its converse are both false after quantification Converse Example as a quantified statement If x is odd, then x is a prime x is a prime, then x is odd false lse converse (false when x = 9) rue when x = 9) (true when x = 2) P Q P Q Q P If x is a prime, then x is odd false Truth table of P Q vs Q P (false when x = 2) P(x) Q(x) P(x) Q(x) Q(x) P(x) (true when x 9) TT T T TF F T x = 9 T T F T FT T F FF T T x = 2 T F T T Lecture 5 8 P Q and Q P cannot be false at the same time.
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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