6 Duality Swap 0 & 1, AND & OR Result: Theorems still true Principle of Duality Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and and + are swapped throughout. Why? Each axiom (A1-A5) has a dual (A1-A5
7 Duality Counterexample: X + X Y = X (T9) X X + Y = X (dual) X + Y = X (T3) ???????????? X + (X Y) = X (T9) X (X + Y) = X (dual) (X X) + (X Y) = X (T8) X + (X Y) = X (T3) parentheses, operator precedence!
8 Dual of a logic expression If F(X 1, X 2, X 3,… Xn, , +, ‘) is a fully parenthesized logic expression involving variables X 1, X 2, X 3,… Xn and the operators +, , and ‘, then the dual of F, written F D, is the same expression with + and swapped. F D (X 1, X 2, X 3,… Xn, +, , ‘)=F(X 1, X 2, X 3,… Xn, , +, ‘)
9 N-variable Theorems Most important: DeMorgan theorems
10 Finite induction Step1: Proving the theorem is true for n=2; Step 2: Proving that if the theorem is true for n=i, then it is also true for n=i+1; Thus the theorem is true for all finite values of n. For example: T12
11 Next… DeMorgan Symbols Representations of logic functions Read Chapter 4.2 and take notes Combinational circuit analysis