28 Consensus Theorem The third term is redundant Can just dropProof in book, but in summaryFor third term to be true, Y & Z both 1Then one of the first two terms must be 1!
29 Complement of a Function Definition:1s & 0s swapped in truth table
30 Truth Table of the Complement of a Function XYZF = X + Y’ZF’1
31 Algebraic Form for Complement Mechanical way to derive algebraic form for the complement of a functionTake the dualRecall: Interchange AND & OR, and 1s & 0sComplement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal)
32 Example: Algebraic form for the complement of a function F = X + Y’ZTo get the complement F’Take dual of right hand sideX . (Y’ + Z)Complement each literal: X’ . (Y + Z’)F’ = X’ . (Y + Z’)
34 From Truth Table to Function Consider a truth tableCan implement Fby taking OR of all terms that correspond to rows for which F is 1“Standard Form” of the function
35 Standard Forms Not necessarily simplest F But it’s mechanical way to go from truth table to functionDefinitions:Product terms – AND ĀBZSum terms – OR X + ĀThis is logical product and sum, not arithmetic
36 Definition: MintermProduct term in which all variables appear once (complemented or not)For the variables X, Y and Z example minterms :X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ
37 Definition: Minterm (continued) Each minterm represents exactly one combination of the binary variables in a truth table.
39 Number of Minterms For n variables, there will be 2n minterms Minterms are labeled from minterm 0, to minterm 2n-1m0 , m1 , m2 , … , m2n-2 , m2n-1For n = 3, we havem0 , m1 , m2 , m3 , m4 , m5 , m6 , m7
40 Definition: MaxtermSum term in which all variables appear once (complemented or not)For the variables X, Y and Z the maxterms are:X+Y+Z , X+Y+Z’ …. , X’+Y’+Z’