204Section22009

204Section22009 - Econ 204 Section 2 GSI: Hui Zheng Key...

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Econ 204 Section 2 GSI: Hui Zheng Key Words Deduction, Contraposition, Contradiction, Induction, Binary Relation, Equivalence Relation, Equivalence Class, Cardinality, Bijection, Numerically Equivalent Section 2.1 Methods of Proof Deduction To prove A ) Z , deduction goes like A ) B ) ±±± ) Y ) Z Contraposition To prove A ) Z , contraposition is to prove : Z ) : A: It goes like : Z ) : Y ) ±±± ) : B ) : A Induction A typical structure of proof is For n = 0 (or other initial value), show that the statement is true. This is the base step. For n = k; suppose that the statement is true. This is the inductive hypothesis. For n = k + 1 ; use what we get from the inductive hypothesis to show that the statement holds for the case of n = k + 1 Conclude that the statement is true for all n . Contradiction To prove A ) Z Z is not true. Then we check whether it leads to results that contradict with A or what we get from A . Contraposition can be regarded as a special case of contradiction. Contraposition means : Z ) : A so that we get results that contradicts with A . A sequence of real numbers f x n g converges to a real number x if 8 " > 0 9 N ( " ) 2 N , for all n > N ( " ) ) j x n ² x j < " . We denote it as x n ! x or lim n !1 x n = x: The following two examples will use this de&nition. We are going to learn the general de&n- ition of convergence of sequences in metric spaces in Lecture 3. However, they are pretty similar. Example 2.1.1 Prove the following statement by deduction: f x n g is a sequence of real numbers. If lim n !1 x n = x > 0 ; then there exists N 2 N such that n > N ) x n > 0 . Solution: Since lim n !1 x n = x > 0 8 " > 0 9 N ( " ) 2 N , for all n > N ( " ) ) j x n ² x j < ": Let " = x 2 > 0 . Then there exists N ( " ) such that n > N ( " ) )j x n ² x j < " = x 2 . Since j x n ² x j <
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.

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204Section22009 - Econ 204 Section 2 GSI: Hui Zheng Key...

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