The Connection Truth Tables to Functions Condition that a is 0 b is 0 c is 1 a b c F 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 a b c a b c 1 a b c 1 0 0 0 1 0 1 1 1 1 0 a b c 1 a b c 1 1 1 0 Function F is true if any of these and terms are true OR F a b c a b c a b c a b c a b c Sum of Products form Seattle Pacific University EE 1210 Logic System Design SOP POS 1 Minterm Shorthand a b c F 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 a b c a b c a b c a b c a b c a b c a b c a b c m0 m1 m2 m3 m4 m5 m6 m7 A minterm has one literal for each input variable either in its normal or complemented form Note Binary ordering A canonical sum of products form of an expression consists only of minterms OR d together F a b c a b c a b c a b c a b c F m1 m2 m3 m5 m6 F m 1 2 3 5 6 Seattle Pacific University EE 1210 Logic System Design SOP POS 2 Minterms of Different Sizes Four variables Two variables a 0 0 1 1 b 0 1 0 1 minterm a b m0 a b m1 a b m2 a b m3 Three variables a 0 0 0 0 1 1 1 1 b 0 0 1 1 0 0 1 1 c 0 1 0 1 0 1 0 1 minterm a b c m0 a b c m1 a b c m2 a b c m3 a b c m4 a b c m5 a b c m6 a b c m7 Seattle Pacific University a 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 b 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 c 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 d 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 EE 1210 Logic System Design minterm a b c d m0 a b c d m1 a b c d m2 a b c d m3 a b c d m4 a b c d m5 a b c d m6 a b c d m7 a b c d m8 a b c d m9 a b c d m10 a b c d m11 a b c d m12 a b c d m13 a b c d m14 a b c d m15 SOP POS 3 Sum of Products Minimization F in canonical sum of products form minterm form F a b c a b c a b c a b c a b c Use algebraic manipulation to make a simpler sum of products form Duplicate term OK Use commutativity to reorder to group similar terms F a b c a b c a b c a b c a b c a b c F a a b c c c a b a a b c Use x x 1 identity F b c a b b c Use distributivity to factor out common terms We will find a better method K maps later Seattle Pacific University EE 1210 Logic System Design SOP POS 4 Product of Sums from a Truth Table A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F 0 0 0 1 1 1 1 1 F 1 1 1 0 0 0 0 0 Find an expression for F the complement F A B C A B C A BC F A B C A B C A BC Complement both sides F A B C A B C A BC F A B C A B C A B C Seattle Pacific University Use DeMorgan s Law to re express as product of sums EE 1210 Logic System Design SOP POS 5 Maxterms A 0 0 0 0 1 1 1 1 B 0 0 1 1 0 0 1 1 C 0 1 0 1 0 1 0 1 F 0 0 0 1 1 1 1 1 F 1 1 1 0 0 0 0 0 F A B C A B C A B C Maxterms To find a Product of Sums form for a truth table Make one maxterm for each row in which the function is zero For each maxterm each variable appears once In its complemented form if it is one in the row In its regular form if it is zero in the row Seattle Pacific University EE 1210 Logic System Design SOP POS 6 Maxterm Shorthand Product of Sums A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 0 1 Maxterms A B C M0 A B C M1 A B C M2 A B C M3 A B C M4 A B C M5 A B C M6 A B C M7 F in canonical maxterm form F A B C A B C A B C F M 0 M1 M 2 F M 0 1 2 Seattle Pacific University EE 1210 Logic System Design SOP POS 7 Boolean operations and gates Theorem Any operation than can be represented by a truth table can be represented in Boolean algebra All truth tables can be made out of only and or and not functions Seattle Pacific University EE 1210 Logic System Design SOP POS 8 NAND NOR expressions Any expression can be made of and ANDs ORs and NOTs We can make ANDs and ORs from NANDs and NORs and NOTs Thus we can make any expression out of NANDs NORs and NOTs We can make NOTs out of a single NAND gate X X So we can make any expression out of just NANDs and NORs note NANDs and NORs are easy to build with switches Seattle Pacific University EE 1210 Logic System Design SOP POS 9 NAND only circuits Using DeMorgan s Law NORs can be made with NANDs We can make any Boolean expression out of only NAND Gates NANDs can be made out of NORs We can make any Boolean expression out of only NOR Gates Seattle Pacific University EE 1210 Logic System Design SOP POS 10 Sum of Products Circuits with NANDs Introduce Double Inverters DeMorgan s Law Seattle Pacific University Sum of Products works well with NANDs EE 1210 Logic System Design SOP POS 11 Product of Sums Circuits with NORs Introduce Double Inverters DeMorgan s Law Product …