Exercise1 A
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Exercise Session 1, Oct 25th ; 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua (Cissy) Chen, Natalia Guseva
Exercise 1 Look at the following pairs of statements and determine if they are logically equivalent. a) ~(p^q); (~p) _ (~q): b) (p ) q)^(q ) r); p ) r: Exercise 2 Negate the following statements: a) For all x, there is a y such that for all z > x, z > y. b) There is a country in which every citizen has a friend who knows all the constellations. Exercise 3 Show that for all n and for all x, 1+x+x2 +x3 +:::+xn =
1 xn+1 1 x .
Exercise 4 Let L(X; Y ) be the vector space of all linear mappings from vector space X to vector space Y . Let T 2 L(Y; Z); S 2 L(X; Y ). Show that the composition of the two linear functions, T (S( )), is linear.
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Exercise 1 Look at the following pairs of statements and determine if they are logically equivalent. a) ~(p^q); (~p) _ (~q): b) (p ) q)^(q ) r); p ) r: Exercise 2 Negate the following statements: a) For all x, there is a y such that for all z > x, z > y. b) There is a country in which every citizen has a friend who knows all the constellations. Exercise 3 Show that for all n and for all x, 1+x+x2 +x3 +:::+xn =
1 xn+1 1 x .
Exercise 4 Let L(X; Y ) be the vector space of all linear mappings from vector space X to vector space Y . Let T 2 L(Y; Z); S 2 L(X; Y ). Show that the composition of the two linear functions, T (S( )), is linear.
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