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Im studying some properties of Boolean Algebra and found the Operation Linkage Laws.

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Im not able to understand how this laws are possible, and the remark on the proof is not really clear for me. Can any one show me the proofs of this laws or share any resources where I can find a more detailed proof of the laws?

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    $\begingroup$ Please transcribe the image as text, to enable textual search. $\endgroup$ – Yuval Filmus Jan 22 '17 at 22:07
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The proposition states that if any of the four statements holds, then the other three also holds. This gives in total 12 different theorems. For the proof, you can use truth tables or the axioms of Boolean algebra.

What the comment is trying to explain is that, say, $xy=x$ isn't universally true (consider $x=0,y=1$). However, if $xy=x$ then also $x+y=y$, and vice versa. This is in contrast to identities such as $x+x=x$, which always hold (for all values of $x$).

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    $\begingroup$ You don't actually have to prove all the 12 implications, though. Usually, when you have a theorem in the form "the following n are equivalent", the smartest way to prove it is to show that (1) implies (2), (2) implies (3), ..., (n) implies (1). $\endgroup$ – quicksort Jan 22 '17 at 22:16

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