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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 03/25/2011.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online