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xwolfhunter
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So I'm reading up on some set theory, and I came to the axiom of pairing. The book uses that axiom to prove/define a set which contains the elements of two sets and only the elements of those two sets. ##~~B## is the set which contains the elements and only the elements of sets ##a## and ##b##.[tex]B=\{x: x=a~\mathsf{or}~x=b\}=\{a,b\}[/tex]
I am hung up on how "or" is used here. I understand why it's formed the way it is, in order to ensure that unambiguously the set ##B## contains exactly the elements of both sets ##a## and ##b##, but the "or" operator confuses me. ##q~\mathsf{or}~z## means "is true if either ##q##,##~z##, or both ##q## and ##z## is true." This means that according to the above, I can't be sure which statement is true (##x=a## or ##x=b##), or whether they're both true, but of course it is implied in the statement that they are both true. I see why they can't use "and" there, because then by necessity ##a=b##, but what is the exact definition of "or" here? It demonstrates different iterations of ##x##, but what are the rules of this and why are both iterations true when it's the "or" operator which is used here?
I just need to know precisely what everything means, sorry :)
I am hung up on how "or" is used here. I understand why it's formed the way it is, in order to ensure that unambiguously the set ##B## contains exactly the elements of both sets ##a## and ##b##, but the "or" operator confuses me. ##q~\mathsf{or}~z## means "is true if either ##q##,##~z##, or both ##q## and ##z## is true." This means that according to the above, I can't be sure which statement is true (##x=a## or ##x=b##), or whether they're both true, but of course it is implied in the statement that they are both true. I see why they can't use "and" there, because then by necessity ##a=b##, but what is the exact definition of "or" here? It demonstrates different iterations of ##x##, but what are the rules of this and why are both iterations true when it's the "or" operator which is used here?
I just need to know precisely what everything means, sorry :)
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