What is Considered a Property of Boolean Algebra?

Question

I was doing some homework for a course I'm taking at uni and came across an equality that I've yet to find anywhere online. I haven't located it on any enumerated lists of the properties of Boolean Algebra. Is this equality in any way significant? Why are certain properties of Boolean Algebra considered significant and others not? The equality is...

$\overline{(A\oplus \overline{B})}=(A\oplus B)$

Proof:

$\overline{(A\oplus \overline{B})} = \overline{((A\bullet B)+(\overline{A}\bullet \overline{B}))} = \overline{(A\bullet B)} \bullet \overline{(\overline{A} \bullet \overline{B})} = (\overline{A} + \overline{B}) \bullet (A + B) = (A\bullet \overline{A})+(\overline{A} \bullet B) + (\overline{B} \bullet A) + (\overline{B} \bullet B) = (\overline{A} \bullet B) + (\overline{B} \bullet A) = A \oplus B$

Answer 1

Since the Boolean statement $\bbox[gold,1pt]{(A\oplus B)}$ is true exactly when $\bbox[tan,2pt]{A\neq B}$, and $\bbox[gold,1pt]{(A\oplus\overline B)}$ exactly when $\bbox[tan,2pt]{A=B}$, therefore the identity $\bbox[gold,1pt]{\overline{(A\oplus\overline B)}=(A\oplus B)}$ holds because: $\bbox[tan,2pt]{\lnot (A=B)\iff (A\neq B)}$.

This is neither surprising nor useful enough to be considered significant the way the far more commonly used de Morgan's Dual Negation Rule is.   But still, keep it in mind.

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Note on symbols:

• $\oplus$ means "exclusive or" ($\veebar$),
• $\bullet$ means "and" ($\land$),
• $+$ means "or" ($\lor$), and
• $\overline{\text{overline}}$ means "not" ($\lnot$).

Answer 2

Partial Answer, about why some properties may be considered important or not. To quote wikipedia about commutativity "The name is needed because there are operations, such as division and subtraction, that do not have it " so in short it may not be listed as a property of boolean algebra because it may be more general.