Ami Pro - LN033011

Ami Pro - LN033011 - Economics 113 UCSD Prof. R. Starr...

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Lecture Notes, March 30, 2011 Mathematical Logic Logical Inference Let A and B be two logical conditions, like A="it's sunny today" and B="the light outside is very bright" A B A implies B, if A then B A B A if and only if B, A implies B and B implies A, A and B are equivalent conditions Proof s Just like in high school geometry. Concept of Proof by contradiction : Suppose we want to show that A B. Ordinarily, we'd like to prove this directly. But it may be easier to show that [not ( A B)] is false. How? Show that [A & (not B)] leads to a contradiction. A: x = 1, B:x+3=4. Then [A & (not B)] leads to the conclusion that 1+3 4 or equivalently 1 1, a contradiction. Hence [A & (not B)] must fail so A B. (Yes, it does feel backwards, like your pocket is being picked, but it works). Set Theory Definition of a Set { } {x | x has property P} {1, 2, . .., 9, 10} = { x | x is an integer, 1 x 10 }. Elements of a set x A ; y A x { x } x { x }   the empty set ( null set), the set with no elements. Subsets if x A x B A B or A B . A A and  A Set Equality A = B if A and B have precisely the same elements A = B if and only if A B and B A Economics 113 Prof. R. Starr UCSD Spring 2011 March 30, 2011 1
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Set Union A B ('or' includes 'and') A B   x x A or x B Set Intersection A B   x x A and x B If we say that A and B are disjoint. A B  Theorem 6.1: Let A, B, C be sets, a. (idempotency) A A A , A A A b. (commutativity) A B B A , A B B A c. (associativity) A   B C    A B   C A   B C    A B   C d. (distributivity) A   B C    A B    A C A   B C    A B    A C Complementation (set subtraction) \ A \ B   x x A , x B Cartesian Product ordered pairs . AxB   x , y x A , y B Note: If x y, then (x, y) (y, x) .
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Ami Pro - LN033011 - Economics 113 UCSD Prof. R. Starr...

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