Lec03_Proof_Techniques

# Lec03_Proof_Techniques - TDS1191 Discrete Structures...

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Discrete Structures 1 [Lecture03][Proof Techniques] TDS1191 Discrete Structures Lecture 03 Proof Techniques Multimedia University Trimester 2, Session 2011/2012

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Proof Techniques – Objectives If a statement is said to be true, then figure out why it is true. Laws of basic algebra, and rules of inference and theorems are used in the proof techniques to prove a statement. Discrete Structures 2 [Lecture03][Proof Techniques] A theorem is a statement that can be shown to be true.
Recall from Lecture 02 P Premises CI Conversion of Implication Add. Addition Simp. Simplification Conj. Conjunction MP Modus Ponens MT Modus Tollens HS Hypothetical Syllogism DS Disjunctive Syllogism Res. Resolution Discrete Structures 3 [p (p q)] q p p q (p q) p [¬q (p q)] ¬p [(p q) ¬p)] q (p) (q) p q [(p q) (¬p r)] (q r) [(p q) (q r)] (p r) [Lecture03][Proof Techniques] Tautologies

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Discrete Structures 4 Proof Techniques 1.Direct Proof 2.Indirect Proof - Proof by Contraposition - Proof by Contradiction 1.Proof by Cases 2.Proofs of Equivalence [Lecture03][Proof Techniques]
Discrete Structures 5 Direct Proof [Lecture03][Proof Techniques] p q can be proved by showing that if p is true, then q must also be true . Step 1: Formalize the facts through variables. Step 2: Represent the statements (premises) using logical expressions. Step 3: Prove that the premises support the conclusion.

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Discrete Structures 6 Direct Proof – Example 1 If the table is full, then it contains a marker. If this entry is the last, then the table is full. Show that if the table contains no marker, then this entry is not the last. [Lecture03][Proof Techniques] Step 3: Show that (p q) (r p) (¬ q ¬r) Formal proof: 1. p q P 2. r p P 3. r q 1, 2, HS 4. ¬q ¬r 3, Contra-positive Solution: Step 1: p : The table is full. q : The table contains marker. r : This entry is the last. Step 2: 1. p q and 2. r p.
Discrete Structures 7 Direct Proof – Example 2 If John misses the bus or wakes up late, he will not reach his office before eight. This morning John woke up late and missed the bus. Show that John did not reach his office before eight. p: John misses the bus. q: John wakes up late. r: John does not reach his office before eight. [(p q) r) (p q)] r Formal proof: 1. (p q) r P 2. p q P 3. p 2, Simp. 4. p q 3, Add. 5. r 1,4, MP (p q) r p q r [Lecture03][Proof Techniques]

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Discrete Structures 8 Direct Proof - Example 3 Instead of having a scenario, question could be set like this as well: Prove that [(¬p q) (¬r p) (r s)] (¬s q) Note: The
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## This note was uploaded on 02/03/2012 for the course IT 1191 taught by Professor Yong during the Spring '11 term at Multimedia University, Cyberjaya.

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Lec03_Proof_Techniques - TDS1191 Discrete Structures...

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