Sum of Products (SOP) & Product of SUMS(POS) SOP & POS A LOGIC CIRCUIT A logic circuit contains one or more basic logic gates. Logic circuits have the properties of Boolean algebra. An AND-OR logic circuit - Called “ Sum of Products”. An OR-AND logic circuit -Called “Product of Sums”. SOP & POS SIMPLIFICATION OF BOOLEAN EXPRESSIONS SOP & POS BOOLEAN ALGEBRA LAWS AND RULES Commutative Law 1. Commutative law of addition: A+B = B+A 2. Commutative law of Multiplication A*B=B*A SOP & POS Associative Law: 1. Associative Law of Addition: A+(B+C)=(A+B)+C 2. Associative Law of Multiplication: A(BC)=(AB)C These laws mean that the grouping of several variables Ored or ANDed together does not matter. SOP & POS Distributive Law: A(B+C)=AB+AC (A+B) . (C+D)= AC+AD+BC+BD (FOIL method) Boolean expression simplification SIMPLIFY Simplify the Boolean function F=AB+ BC + B′C. Solution. F = AB + BC + B′C = AB + C(B + B′) = AB + C Boolean expression simplification SIMPLIFY Simplify the Boolean function F= A′B′C + A′BC + AB′. Solution. F = A′B′C + A′BC + AB′ = A′C (B′+B) + AB′ = A′C + AB′ SOP & POS SIMPLIFY F= ABC’ + AB’C + A’BC + ABC [Since X+X = X, ABC can be repeated many times] = ABC’ + ABC + AB’C + ABC + A’BC+ABC = AB(C’+C) + AC(B’+B) + BC(A’+A) = AB+ AC+BC Simplify : A’B’C+ A’BC +AB’ Karnaugh Maps (K-Maps) SOP & POS Karnaugh map is a method used to simplify Boolean expressions. The Boolean expressions must be in SOP format before they can be put into a map. We can transfer logic values from a Boolean statement or a truth table into a Karnaugh map SOP & POS It contains boxes called cells Each cell represents one of the 2n possible products that can be formed from n variables Two variable map contains 4 cells (22), 3 variable map contains 8 cells(23), 4 variable map contains 16 cells(24) 2 Variable cells A B 0 1 0 00 A’B’ 0 01 A’B 1 1 10 AB’ 2 11 AB 3 SOP & POS 3 Variable cells C AB 00 11 10 010 110 A’B’C’ 0 A’BC’ 2 ABC’ 6 100 4 AB’C’ 001 011 111 101 A’B’C 1 A’BC 3 ABC 7 AB’C 5 000 0 01 1 SOP & POS The Karnaugh map is completed by entering a '1‘(or ‘0’) in each of the appropriate cells Within the map, adjacent cells containing 1's (or 0’s) are grouped together in twos, fours, or eights SOP & POS SOP & POS SOP & POS SOP & POS SOP & POS Simplify the Boolean function F = A′BC + A′BC′ + AB′C′ + AB′C. A BC 00 01 11 10 0 1 1 1 = A’B + AB’ 1 1 SOP & POS Simplify the Boolean function F (A, B, C) = Σ (0, 2, 4, 5, 6). A BC 00 01 11 10 0 1 1 1 1 =AB’+C’ 1 1