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It is more like or not. The following dictionary entry explicitly gives the usage:
安否=safe or not, 可否=possible or not, 合否=(about exams) ...
When you look at the dictionary meaning of 安 in Japanese, you can see that while it does mean cheap (安い/やすい) it can also mean peaceful. I'd also like to note that many Japanese words have two readings,...
The 関 entry in Kotobank is backed by a version of the 日本国語大辞典【にほんこくごだいじてん】 that doesn't include as much detail as my local copy, which includes this bit about the etymology: