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Lec02_Formal_Reasoning

Lec02_Formal_Reasoning - TDS1191 Discrete Structures...

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Discrete Structures 1 [Lecture02][Formal Reasoning] TDS1191 Discrete Structures Lecture 02 Formal Reasoning Multimedia University Trimester 2, Session 2011/2012

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Discrete Structures 2 Reasoning: Cognitive process of looking for reasons, beliefs, and conclusions Mathematics way: sequence of statements (arguments), with proper reasoning rules (rules of inference), give us clues to draw a valid conclusion. Reasoning: Cognitive process of looking for reasons, beliefs, and conclusions Mathematics way: sequence of statements (arguments), with proper reasoning rules (rules of inference), give us clues to draw a valid conclusion. Formal Reasoning [Lecture02][Formal Reasoning]
Discrete Structures 3 I am in the Discrete Structures course in MMU. After doing some research of the last year courses I found the following facts: - Students study hard or don’t study hard for this subject. - If the student studies hard, he will pass the course. - If the student does not study hard, he will not pass the course. Hey man, you are also in this course. I think you will only pass this course, if you study hard. I am in the Discrete Structures course in MMU. After doing some research of the last year courses I found the following facts: - Students study hard or don’t study hard for this subject. - If the student studies hard, he will pass the course. - If the student does not study hard, he will not pass the course. Hey man, you are also in this course. I think you will only pass this course, if you study hard. Assume that a friend tells you the following: Formal Reasoning: Warm-up Exercise Is your friend right? Should you follow his advice? FYI: The example is made extremely simple on purpose! [Lecture02][Formal Reasoning]

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Discrete Structures 4 First, we formalize the facts spoken in English: [Lecture02][Formal Reasoning] Let’s define p: The student studies hard. q: The student will pass the course. - Students study hard or don’t study hard for this subject. - If the student studies hard, he will pass the course. - If the student does not study hard, he will not pass the course. You will only pass this course, if you study hard. p ¬p p q p q ¬p ¬q An argument in propositional logic is a sequence of propositions where all but the last one are called premises and the last one is called conclusion . Formal Reasoning: Mathematical Proof
Discrete Structures 5 Now, we prove that the premises support the conclusion. We have to show that (p ¬p) (p q) (¬p ¬q) (p q) is a tautology and if so, we write (p ¬p) (p q) (¬p ¬q) (p q) We say that the premises p ¬p, p q and ¬p ¬q imply the conclusion p ↔ q [Lecture02][Formal Reasoning] Formal Reasoning: Mathematical Proof

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Method 1: Use logical equivalences If all the given premises imply the conclusion, then we shall get a tautology. (p
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Lec02_Formal_Reasoning - TDS1191 Discrete Structures...

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