I'm not sure I understand. Suppose A has a segment inherited from his mother, who inherited it from her father (C say). B matches this segment, inherited from his father, who inherited it from his father, also C. A and B are first cousins and match with a DNA segment they each inherited from their common grandfather. How is this a false positive?
[/quote]
It's not. But testing companies can't physically phase your DNA, so they have no way of knowing which of the values obtained on any given segment have been inherited from your mother and which from your father. Imagine that another individual who is not related to you at all, has a match against a specific segment of your DNA, but the segments only appear identical because the unphased data matches. In reality, neither that persons maternal or paternal values match yours, but the combination of the two jumbled together does. The apparent match is just a random chance.
The shorter the apparently matching segments are, the greater the possibility that they aren't genuine matches at all. Roughly 50% of "matching" 8cM segments may not be true genetic matches, juts a chance combination that happens to appear identical in an unrelated person, due to the limitations of testing. The smaller the matching segment, the greater the probability that it is a false positive, but the larger the match, the more likely it is genuine. Anything over 15cM is generally considered safe.
I might not have explained that very well, so have a look at the illustrations here, particularly the false positive or identical by chance match:
https://dna-explained.com/2017/01/19/concepts-segment-size-legitimate-and-false-matches/