Duality principle in boolean algebra
Olivia Zamora 
All the definitions I came across so far stated, that if a statement is true, then also its dual statement is true and this dual statement is obtained by changing + for ., 0 for 1 and vice versa.
However when I say 1+1, whose dual statement according to the above is 0.0, I get opposite results, that is:
1 + 1 = 1
0 . 0 = 0How should I understand this duality principle ?
$\endgroup$6 Answers
$\begingroup$"$1 + 1 = 1$" is a statement (a boolean statement, in fact), and indeed, $1 + 1 = 1$ happens to be a true statement.
Likewise, the entire statement "$0 \cdot 0 = 0$" is a true statement, since $0 \cdot 0$ correctly evaluates to false: and this is exactly what "$0 \cdot 0 = 0$" asserts, so it is a correct (true) statement about the falsity of $0 \cdot 0$.
The duality principle ensures that "if we exchange every symbol by its dual in a formula, we get the dual result".
- Everywhere we see 1, change to 0.
- Everywhere we see 0, change to 1.
- Similarly, + to $\cdot$, and $\cdot$ to +.
More examples:
(a) 0 . 1 = 0: is a true statement asserting that "false and true evaluates to false"
(b) 1 + 0 = 1: is the dual of (a): it is a true statement asserting that "true or false evaluates true."
(c) 1 . 1 = 1: it is a true statement asserting that "true and true evaluates to true".
(d) 0 + 0 = 0: (d) is the dual of (c): it is a true statement asserting, correctly, that "false or false evaluates to false".
The statement is the full equation, including the = sign. 1+1 is neither true nor false: it takes the value 1, but it is not actually saying anything. Analogously, the expression "Tom has a cat" is neither true nor false (without specifying who Tom is) - it is an expression which could be true or false, depending on who we mean when we say "Tom".
On the other hand, the statement 1+1=0 is a false. Analogously, the statement "If Tom has a cat then Tom has no cats" is false, no matter who we mean when we say "Tom".
In this case, 1+1=1 is the true statement. Its dual is 0.0=0, which is also a true statement.
The statement is not 1+1 but rather 1+1=1.
What the duality principle says is that "if you exchange every symbol by its dual in a formula you get the dual result".
$\endgroup$ $\begingroup$The difference between an expression and a statement is that statement is like an equation and expression is like variable.
"Duality of a statement is true" - What this means is that when you equate the dual of both the expressions in LHS and RHS of a statement(equation), the statement still makes sense.
Observe that in your example both 1+1 = 1 & 0.0 = 0 ,which are dual to each other, are true statements.
$\endgroup$ $\begingroup$The dual of $1.0$ which is $0+1$, will be obtained by interchanging $AND$ to $OR$ and $1$'s to $0$'s. The result of those function needn't be same.The results will be opposite to each other $1.0=0;0+1=1$, but they are the dual form of the single function $1.0$.
For example, the dual function of $XNOR$ is $XOR$.
In my humble opinion, it is more helpful to describe the duality relationship in terms of variables. Example: Expression: x + x = x Dual of expression: x * x = x x could be 1 or 0. In all iterations, the expressions are correct. Good Luck!
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