Homework 1 - Solutions

Homework 1 - Solutions - UCLA Mathematics 115A: Homework #1...

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UCLA Mathematics 115A: Homework #1 Solutions Summer Session C, 2010 August 15, 2010 Problems from the supplementary section: Problem 2.3: The sentence is equivalent to the statement: ( a R ) P ( a,x ) Q ( x ) . (1) However if a = 0 then ax = 0 for all x R ; therefore the statement Q ( x ) may be false when P ( a,x ) is true! We can chance the statement (1) to: ( a R ) P ( a,x ) Q ( x ) . (2) This is true since whenever a 6 = 0, ax = 0 implies x = 0, since in this case a has a multiplicative inverse in R . Problem 2.4: I’ve translated all the statements in the problem into mathematical notation: a) ( x A )( b B )( b > x ) . b) ( x A )( b B )( b > x ) . c) ( x,y R )( f ( x ) = f ( y )) ( x = y ) . This is equivalent to the statement: ( x,y R )[ ¬ ( f ( x ) = f ( y ))] ( x = y ), since ( A B ) ( ¬ A B ). d) ( b R )( x R )( f ( x ) = b ). e) ( x,y R )( ± P )( δ P )( | x - y | < δ ) ( | f ( x ) - f ( y ) | < ± ) . This is equivalent to ( x,y R )( ± P )( δ P )[ ¬ ( | x - y | < δ )] ( | f ( x ) - f ( y ) | < ± ) for the same reason as in part c). f) ( ± P )( δ P )( x,y R )( | x - y | < δ ) ( | f ( x ) - f ( y ) | < ± ). Again, this is equivalent to ( ± P )( δ P )( x,y R )[ ¬ ( | x - y | < δ )] ( | f ( x ) - f ( y ) | < ± ). Now you can simply negate the statements and translate into English: a) ¬ [( x A )( b B )( b > x )] [( x A )( b B )( b x )]. In English: “There exists an x in A so that for all b in B , b is less than or equal to x .” b) Similarly: “For all x in A there exists b in B such that b is less than or equal to x . ” c) “ There exists x,y in R such that f ( x ) = f ( y ) and x 6 = y .” 1
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d) “There exists a b in R such that for all x in R f ( x ) 6 = b . ” e) “There exist x,y R such that there exists an ± P so that for all δ P , | x - y | < δ and | f ( x ) - f ( y ) | ≥ ± f) “There exists an ± P such that for all δ P there exist x,y R such that | x - y | < δ and | f ( x ) -
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This note was uploaded on 02/02/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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Homework 1 - Solutions - UCLA Mathematics 115A: Homework #1...

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