tag:blogger.com,1999:blog-5566281.post5207321991036773369..comments2015-12-06T08:32:53.118+01:00Comments on Jurjan-Paul's blog: OverlapJurjan-Paul Medemahttp://www.blogger.com/profile/17311716656988176266noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-5566281.post-4423399741219975582009-02-01T00:47:00.000+01:002009-02-01T00:47:00.000+01:00What you have proved is "if [a1, a2] and [b1,...What you have proved is "if [a1, a2] and [b1, b2] overlap then a1 <= b2 ∧ a2 >= b1". But for an "if and only if", you also need to prove it the other way, i.e that if a1 <= b2 ∧ a2 >= b1 then [a1, a2] and [b1, b2] overlap. So assume that a1 <= b2 ∧ a2 >= b1 and show that it necessarily follows that there must be at least one element in both [a1,a2] and [b1,b2].<BR/> <BR/>To do so, you could consider the two numbers a1 and b1. One of the following statements must be true:<BR/>1) a1 = b1<BR/>2) a1 < b1<BR/>3) a1 > b1<BR/><BR/>If a1 = b1, then we are done because a1 is in both ranges.<BR/><BR/>If a1 < b1, recall we have assumed that a2 >= b1. So <BR/>a1< b1 <= a2. Thus b1 is in [a1,a2] and so is in both ranges.<BR/><BR/>If a1 > b1, recall we have assumed that a1 <= b2. So b1 < a1 <= b2. Thus a1 is in [b1,b2] and so is in both ranges.<BR/><BR/>For each possibility we have shown that it follows that there exists at least one one element in both [a1,a2] and [b1,b2]. Q.E.D.<BR/><BR/>I disagree that you should change the <= to <<BR/>I would consider a1=b1 to be an actual overlap - the ranges would then overlap at a point.Anonymousnoreply@blogger.com