204Section32009 - Econ 204 Section 3 GSI: Hui Zheng Key...

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Econ 204 Section 3 GSI: Hui Zheng Key Words Supremum Property, Intermediate Value Theorem. Section 3.1 Field and Vector Space Lecture 2 Section 1.5 Field Axioms (nine properties). Lecture 2 Section 1.5 Vector Space Axioms (eight properties). Example 3.1.1 Show that Q ( p 2) over Q is a vector space. Recall in lecture Q ( p 2) = f q + r p 2 : q; r 2 Q g . Check the eight properties. Associativity of + : 8 q 1 + r 1 p 2 ; q 2 + r 2 p 2 ; q 3 + r 3 p 2 2 Q ( p 2) ; ( q 1 + r 1 p 2+ q 2 + r 2 p 2)+ q 3 + r 3 p 2 = q 1 + r 1 p 2+( q 2 + r 2 p 2+ q 3 + r 3 p 2) Commutativity of + : 8 q 1 + r 1 p 2 ; q 2 + r 2 p 2 2 Q ( p 2) ; q 1 + r 1 p 2 + q 2 + r 2 p 2 = q 2 + r 2 p 2 + q 1 + r 1 p 2 Existence of vector additive identity: 9 0 = 0 + 0 p 2 2 Q ( p 2) ; 8 q 1 + r 1 p 2 2 Q ( p 2) ; q 1 + r 1 p 2 + 0 = q 1 + r 1 p 2 Existence of vector additive inverse: 8 q + r p 2 2 Q ( p 2) ; 9 ( ± q ) + ( ± r ) p 2 2 Q ( p 2) ; s:t: q + r p 2 + ( ± q + ( ± r ) p 2) = 0 Distributivity of scalar multiplication over vector addition: 8 a 2 Q ; 8 q 1 + r 1 p 2 ; q 2 + r 2 p 2 2 Q ( p 2) ; a ² ( q 1 + r 1 p 2+ q 2 + r 2 p 2) = a ² ( q 1 + r 1 p 2)+ a ² ( q 2 + r 2 p 2) Distributivity of scalar multiplication over scalar addition: 8 a; b 2 Q ; 8 q + r p 2 2 Q ( p 2) ; ( a + b ) ² ( q + r p 2) = a ² ( q + r p 2) + b ² ( q + r p 2) Associativity of ² : 8 a; b 2 Q ; 8 q + r p 2 2 Q ( p 2) ; ( a ² b ) ² ( q + r p 2) = a ² f b ² ( q + r p 2) g Multiplicative identity: 8 q + r p 2 2 Q ( p 2) ; 9 1 2 Q ; 1 ² ( q + r p 2) = q + r p 2 X ³ R . We say u is an upper bound for X if 8 x 2 X x ´ u and l is a lower bound for X if 8 x 2 X x µ l . X is bounded above. The supremum of X , written sup X; is the smallest upper bound for X , i.e. sup X 8 x 2 X x ´ sup X and 8 y < sup X 9 x 2 X x > y: The of X , written inf X; is the largest lower bound for X , i.e. inf X 8 x 2 X x µ inf X and 8 y > inf X 9 x 2 X x < y: Example 3.2.1 Sup and Inf Suppose X = f 1 x j 1 ´ x < + 1g sup X = 1 and inf X = 0 .
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204Section32009 - Econ 204 Section 3 GSI: Hui Zheng Key...

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