Question

prove the following: there exist integers x < 100 and y < 30 such that x + y < 128 and for all real numbers r and s, if r > x and s > y, then (r-50)(s-20) > 390.

This is an intro to prrof writing so please, if you help, show the logic form with quantifiers of the initial statement and the form for the proof you will use. TY

Answer 1

can I just say y = 29 and x = 99 there for (49)(9) > 390 so if r > x and s > y then (r-50)(s-20) > 390? something like that.....

Answer 2

Something like that. However, you want \(x+y<128\) so I would use \(x=98\), and \(y=29\). Once you have that, show that \((98-50)(29-20)>390\).

Answer 3

word ty

Answer 4

This means, that if you choose \(r>98\) and \(s>29\), you know that\[(r-50)(s-20)>(98-50)(29-20)>390\]

Answer 5

You're welcome.