Talk:Boolean values: Difference between revisions
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:::::: Quoting that page: ''There are at least four different and incompatible systems of notation for Boolean rings and algebras. ... The old terminology was to use ... "Boolean algebra" to mean a Boolean ring with an identity.'' --[[User:Rdm|Rdm]] 20:03, 1 May 2012 (UTC)
::::::: Yes, and read on: ''Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it'' --[[User:Ledrug|Ledrug]] 20:06, 1 May 2012 (UTC)
:::::::: Except: that statement is provably false. Consider the ring I have been using as an example:
::::::::: Er, no: 0 is a lousy identity for GCD. What's GCD(0, 0)? --[[User:Ledrug|Ledrug]] 20:34, 1 May 2012 (UTC)
:::::::::: You are correct that it's the wrong identity. So I fixed my previous statement.
:::::::::: That said, GCD maps to Logical OR, and LCM maps to Logical AND when 0 maps to false and 1 maps to true. GCD(0, 0) is AND(false, false). Note that we are defining LCM and GCD to satisfy the constraints of Boolean Algebra (in the sense of a Boolean Ring with Identity), and this drives the definition in the cases which would otherwise be undefined. --[[User:Rdm|Rdm]] 20:47, 1 May 2012 (UTC)
::: As to Shannon "dumbing it down" (I guess you'd rather put it this way) to two values, again, since it's about decisions, making the results always "yes" or "no" is at least practical.
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