Mathematical Thinking: Problem-Solving and Proofs (2nd Edition)

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2.3. In symbols, the sentence is “( a R )( x R ) P ( a,x ) Q ( x ).” This is false since P (0 , 1) is true, but Q (1) is false. One way to change the quantiﬁers to make a true statement is “( a R - { 0 } )( x R ) P ( a,x ) Q ( x ).” Another way is “( a R )( x R ) P ( a,x ) Q ( x ).” 2.4. (a) There exists an x A such that for all b B , b x . (b) For each x A , there is some b B such that b x . (c) There is an x R and a y R such that x 6 = y , but f ( x ) = f ( y ). (d) There is some b R such that, given any x R , f ( x ) 6 = b . (e) One can ﬁnd an x R , a y R , and an ± P such that for every δ P , | x - y | < δ and | f ( x ) - f ( y ) | ≥ ± . (f) There exists an ± P such that for every δ P , there are x,y R where δ P , | x - y | < δ and | f ( x ) - f ( y ) | ≥ ± . 2.44.

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HW4solns - 2.3. In symbols, the sentence is "(a R)(x R)P...

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