While I try to understand the \"Availability\" (A) and \"Partition tolerance\" (P) in CAP, I found it difficult to understand the explanations from various articles.
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Considering P in equal terms with C and A is a bit of a mistake, rather '2 out of 3' notion among C,A,P is misleading. The succinct way I would explain CAP theorem is, "In a distributed data store, at the time of network partition you have to chose either Consistency or Availability and cannot get both". Newer NoSQL systems are trying to focus on Availability while traditional ACID databases had a higher focus on Consistency.
You really cannot choose CA, network partition is not something anyone would like to have, it is just an undesirable reality of a distributed system, networks can fail. Question is what trade off do you pick for your application when that happens. This article from the man who first formulated that term seems to explain this very clearly.