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Unformatted text preview: ELEN 601: Exam 1 On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Signature: Name: October 3, 2007 You may use the following list of mathematical expressions. Definition A metric on a set X is a function d : X X R that satisfies the following properties, 1. d ( x,y ) x,y X ; equality holds if and only if x = y 2. d ( x,y ) = d ( y,x ) x,y X 3. d ( x,y ) + d ( y,z ) d ( x,z ) x,y,z X . Definition Let X and Y be topological spaces. A function f : X Y is continuous if for each open subset O Y , the set f 1 ( O ) is an open subset of X . Theorem The sequence x 1 ,x 2 ,... converges to x in a metric space if, given &gt; , there exists M N such that m &gt; M implies d ( x m ,x ) &lt; . Theorem Let f : X Y where X is a metrizable space. The function f is continuous if and only if for every convergent sequence x n x in X , the sequence f ( x n ) converges to f ( x ) . Definition Let v 1 ,...,v n be an ordered basis for the n-dimensional vector space V . Every vector w V can be expressed uniquely as w = n summationdisplay i =1 s i v i . Definition Let V be a vector space over F . A subspace of V is a subset W V which is itself a vector space over F . Theorem A non-empty subset W V is a subspace of V if and only if for every pair of vector w 1 ,w 2 W and every scalar s F the vector sw 1 + w 2 is again in W . Theorem If x , y R n and c R , with x y = x 1 y 1 + x n y n . Then, the following inequalities hold | x y | x x y y = bardbl x bardbl 2 bardbl y bardbl 2 and bardbl x + y bardbl 2 bardbl x bardbl 2 + bardbl y bardbl 2 . 1 Problems: 1. Short answer questions: (a) 1 pt Given a conditional statement of the form P Q , what is the name of the related statement Q P ? Contrapositive. (b) 1 pt If the statement P Q is a tautology, what is the name of the meta-statement that characterizes P and Q ? Implication, P Q . (c) 1 pt Negate the statement: ( x )( y )[ L ( x,y ) M ( x,y )] . ( x )( y )[ L ( x,y ) M ( x,y )] 2. Assign one word to each of the following sentences: conjunction, disjunction, contradiction, fallacy, equivalence, existential quantifier, basis, topology, limit points of A , closure of A , interior of A , complement of A . (a) 0.5 pt A component of an argument that is demonstrably flawed in its logic or form, thus rendering the argument invalid. Fallacy. (b) 0.5 pt This meta-statement asserts that the statement P Q is a tautology....
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- Fall '08
- Topology, Metric space, Open set, Topological space, g Ve