lecture03 - 15-251 Great Theoretical Ideas in Computer...

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1 15-251 Great Theoretical Ideas in Computer Science Proofs and Logic Lecture 3 (September 6, 2011) P, P Q Q 1+1 = 2 do we need a proof? In mathematics, sometimes your intuition can be dead wrong. We know that 1 2 A solid ball in 3-dimensions can be cut up into a finite number of pieces, so that these pieces can be moved around and assembled into two identical copies of the original ball. Not possible in 1 or 2 dimensions, btw. So it really pays off to: Formalize concepts, give precise definitions Make implicit assumptions explicit Write careful proofs, where every step can be checked carefully. Even “mechanically” using a “computing machine”, if you will What is a proof?
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2 3. 3. Math. and Logic . A sequence of steps by which a theorem or other statement is derived from given premises. 1. a. 1. a. Something that proves a statement; evidence or argument establishing a fact or the truth of anything, or belief in the certainty of something; an instance of this. Let’s ask the OED What is a proof? Intuitively, a proof is a sequence of “statements”, each of which follows “logically” from some of the previous steps. What are “statements”? What does it mean for one to follow “logically” from another? We’ll lay the groundwork for these questions in today’s lecture… But we will come back to these questions again later… Propositions A proposition is a statement that is either true or false. snow is white 2+2 = 5 Socrates had six digits on his left hand the summer solstice occurs on June 21 st Propositions A proposition is a statement that is either true or false. snow 2+2 is xadk keosign sziable? these are not propositions Complex Statements If snow is white then the sun rises in the west Socrates was bald and Cicero ate a pie 2 + 2 = 5 if and only if 4 + 4 = 9 Snow is not white
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3 Complex Statements snow is white the sun rises in the west Socrates was bald Cicero ate a pie 2 + 2 = 5 4 + 4 = 9 ¬ (Snow is white) Well-Formed Statements Every “simple” proposition is a well-formed propositional statement If A and B are well-formed so are (A B), (A B), ¬ A, (A B), (A B) and or not if… then if and only if The “meaning” of these connectives (A B) “A and B” (A B) “A or B” ¬ A “not A” true if both A and B are true true if at least one of A and B is true true if A is false A B A Æ B T T T T F F F T F F F F A B A B T T T T F T F T T F F F A ¬ A T F F T The “meaning” of these connectives (A B) “if A then B” what are the rules for this? “A implies B”
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This note was uploaded on 11/03/2011 for the course CS 251 taught by Professor Gupta during the Spring '11 term at Carnegie Mellon.

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lecture03 - 15-251 Great Theoretical Ideas in Computer...

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