Shannon's inversion theorem

The inversion rate of Shannon is a significant, by Claude Shannon derived sentence in the Boolean algebra , which is a generalization of De Morgan's laws is for algebraic expressions with Boolean variables. Thus, the inversion of any such expression, as long as it is formed using only the operators ∧, ∨, and ¬ (i.e. conjunction, disjunction, and negation), can be achieved by swapping ∧ and ∨ and negating each literal. Literals are understood as Boolean quantities A, B, C, ... as well as ¬A, ¬B, ¬C, ..., i.e. as simple or negated quantities that are not combined.

Formulation of the sentence

Be a Boolean function on literals and the operator tuple . Then applies ${\ displaystyle f \ colon X ^ {n} \ rightarrow X}$ ${\ displaystyle n}$ ${\ displaystyle x_ {1}, ..., x_ {n}}$ ${\ displaystyle (\ lor, \ land)}$

{\ displaystyle {\ overline {f (x_ {1}, x_ {2}, ..., x_ {n}, \ lor, \ land)}} = f ({\ overline {x_ {1}}}, {\ overline {x_ {2}}}, ..., {\ overline {x_ {n}}}, \ land, \ lor)}

Individual evidence

^ Roland Woitowitz, Klaus Urbanski: Digital technology - a textbook and exercise book . 5th, revised and expanded edition. Springer Verlag, 2007, ISBN 978-3-540-73672-1 , pp. 31 .

[1] Roland Woitowitz, Klaus Urbanski: Digital technology - a textbook and exercise book . 5th, revised and expanded edition. Springer Verlag, 2007, ISBN 978-3-540-73672-1 , pp. 31 .