Three Representations of Logic Functions Logic Functions of 2 Variables AND OR NOT X YF0F1F2F3F4F5F6F7 0 000000000 1. Logic Expression X.Y X+Y X’ X 0 100001111 ~X !X 1 000110011 2. Truth Table XYX.Y XYX+Y 1 101010101 00 0 00 0 XX' X Y F8 F9 F10 F11 F12 F13 F14 F15 0 011111111 01 0 01 1 01 0 100001111 1 0 0 1 0 1 1 0 101 0 0 0 1 1 0 0 1 1 11 1 11 1 1 101010101 3. Circuit Diagram • F1 is called a logical AND, denoted by X.Y / Schematic • F6 is called an XOR (Exclusive-OR), denoted by X ⊕ Y • F7 is called OR, denoted by X + Y • F8 is NOR, denoted by X + Y • F9 is called an XNOR (Exclusive-NOR), denoted by X ⊕ Y • F14 is NAND, denoted by X.Y 1 2 Truth Tables for 2 Variable Functions Which Truth Tables Are the Same? 1)BAF 2) ABF X Y AND X Y OR X Y XOR 000 000 0 0 0 0 0 0 0 0 0 010 010 0 1 0 0 1 1 0 1 1 101 101 1 0 0 1 0 1 1 0 1 110 110 1 1 1 1 1 1 1 1 0 X Y NAND X Y NOR X Y XNOR 0 0 1 0 0 1 0 0 1 3) BAF 0 1 1 0 1 0 0 1 0 110 1 0 1 1 0 0 1 0 0 101 1 1 0 1 1 0 1 1 1 010 000 3 4 Logic Gate Symbols Half and Full Adder Truth Tables • AND denoted by X.Y • NAND denoted by X.Y 1)BAS 2) ABC 000 000 011 010 101 100 110 111 • OR denoted by X + Y • NOR denoted by X + Y CBAS CBAC 0 0 0 0 0 0 0 0 • XOR denoted by X ⊕ Y • XNOR denoted by X ⊕ Y 0011 0010 0101 0100 0110 0111 • NOT denoted by X’ or X 1001 1000 1010 1011 1100 1101 1111 1111 5 6 Truth Table with Three Inputs Truth Table with Four Inputs XYZW F • Three inputs X, Y, and Z; Output is F • Four inputs X, Y, Z, and W; Output is F 0000 0 • Logic Function: • Logic Function: 0001 0 F = 1 if and only if there is a 0 to the left of a 1 in the input F = 1 if and only if number of variables 0010 0 • Truth Table: with value 1 is more than the number of 0011 0 X Y Z F Min term variables with value 0 0100 0 0 0 0 0 • Truth Table: 0101 0 0 0 1 1 0110 0 0 1 0 1 0111 1 0 1 1 1 • Logic Expression: 1000 0 1 0 0 0 F= 1 0 1 1 1001 0 1 1 0 0 1010 0 1 1 1 0 1011 1 1100 0 • Logic Expression: 1101 1 F= 1110 1 1111 1 7 8 The Idea of Min Term / Product Term Truth Table with Two Inputs XY F A B X Y Min Term • Two inputs X and Y; Output is F 00 1 1 0 0 0 X' Y' • Logic Function: 0 1 X' Y 01 0 0 0 F = 1 if and only if X = Y 1 0 X Y' • Truth Table: 10 0 0 0 1 1 X Y XY F Min Term 11 1 0 1 00 1 X’Y’ 01 0 X’Y • Each row in a truth table represents a unique combination of 10 0 XY’ variables 11 1 XY • Each row can be expressed as a logic combination specifying • Logic Expression: when that row combination is equal to a 1 F=X’Y’.1 + X’Y.0 + XY’.0 + XY.1 • The term is called a MIN TERM or a PRODUCT TERM =X’Y’ + XY • Thus F = A + B = X’Y’ + XY 9 10 Min / Product terms for more variables Multiplexer Circuit XYZW Min Term s f (s, x , x ) XYZ Min Term sx x f (s, x , x ) 1 2 1 2 1 2 x 0000 X' Y' Z' W' 0 1 0 0 0 0 000 X' Y' Z' 0001 X' Y' Z' W 1 x2 001 X' Y' Z 0010 X' Y' Z W' 0 0 1 0 010 X' Y Z' 0011 X' Y' Z W 0 1 0 1 More compact truth-table representation 011 X' Y Z 0100 X' Y Z' W' 0 1 1 1 0101 X' Y Z' W 100 X Y' Z' 1 0 0 0 101 X Y' Z 0110 X' Y Z W' f=s’xf = s’.x1 +sx+ s.x2 0111 X' Y Z W 1 0 1 1 110 X Y Z' x 1000 X Y' Z' W' 1 s 111 X Y Z 1 1 0 0 1001 X Y' Z' W 1 1 1 1 1010 X Y' Z W' f x1 0 f 1011 X Y' Z W Truth Table x s 2 1 1100 X Y Z' W' x 1101 X Y Z' W 2 1110 X Y Z W' Circuit Graphical symbol 1111 X Y Z W 11 12 Canonical Sum-of-Product Expression Shorthand Notation for Canonical SOP • A product (min) term is a unique combination of variables: • We can also assign an integer to represent each input – It has a value of 1 for only one input combination combination – It is 0 for all the other combinations of variables • Thus the function produces a 1 for input combinations 2, 4, 7 • To write an expression, we need not write the entire truth table • Therefore, the function can be written as f(x,y,z)= ∑m(2,4,7) • We only need those combinations for which function output is 1 Index for • For example, for the function below: f = x’yz’+xy’z’+xyz x y z f shorthand x y z f Min term notation 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 x’yz’ 0 1 0 1 2 0 1 1 0 0 1 1 0 3 1 0 0 1 xy’z’ 1 0 0 1 4 1 0 1 0 1 0 1 0 5 1 1 0 0 1 1 0 0 6 1 1 1 1 7 1 1 1 1 xyz • This is called the Canonical Sum-of-Product (SOP) Expression 13 14 Max / Sum Terms Canonical Product-of-Sum Expression • A max (sum) term is also a unique combination of variables • A function can also be written in terms of max terms – However, it is opposite of a min term • The function is product of all max terms for which function is 0 – It has a value of 0 for only one input combination • For example, the same function of three variable x, y, and z – It is 1 for all the other combinations of variables produces 0 for xyz=000, 011, 101, then – That is why it is called a max (sum) term – F = (x+y+z).(x+y’+z’).(x’+y+z’) – Each row in truth table has a max term corresponding to it • This is called the Canonical Product-of-Sum (POS) Expression • Example, a max term (x+y+z) is 0 for combination xyz=000 only • The function can also be written as F(x, y, z)= ∏ M (0,3,5) X Y Z F F’ X Y Max Term 0 0 0 0 1 F = (F’)’ 0 0 X + Y 0 0 1 1 0 = (x’y’z’ + x’yz + xy’z)’ 0 1 X + Y’ 0 1 0 1 0 = (x+y+z).(x+y’+z’).(x’+y+z’) 1 0 X’ + Y 0 1 1 0 1 1 1 X’ + Y’ 1 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 1 0 15 16 Truth Tables and Logic Expression for Adder Multiple Forms and Equivalence A B C X Y • Canonical Sum-of-Product form X (A, B,C) = m(1,2,4,7) 0 0 0 0 0 ∑ • Canonical Product-of-sum form 0 0 1 1 0 Y(A, B,C) = ∑m(3,5,6,7) • How to convert one from other? 0 1 0 1 0 • Minterm expansion of X to minterm expansion of X’ 0 1 1 0 1 – Just take the terms that are missing 1 0 0 1 0 X (A,B,C) = ∑m(1,2,4,7) X '(A, B,C) = ∑m( ) 1 0 1 0 1 X (A,B,C) = ∏M ( ) 1 1 0 0 1 Y(A, B,C) = M ( ) • Maxterm expansion of X to maxterm expansion of X’ 1 1 1 1 1 ∏ – Just take the terms that are missing X = A’B’C + A’BC’ + AB’C’ + ABC X (A, B,C) = ∏M ( ) X '(A,B,C) = ∏M ( ) Y = A’BC + AB’C + ABC’ + ABC 17 18 Boolean Algebra Axioms of Boolean Algebra • An algebraic structure consists of • 1a: 0.0 = 0 – a set of elements {0, 1} – binary operators {+, .} 1b: 1+1 = 1 – and a unary operator { ’ } • 2a: 1.1 = 1 2b: 0+0 = 0 • Introduced by George Boole in 1854 • 3a:01=10=03a: 0.1 = 1.0 = 0 • An effective means of describing circuits built with switches 3b: 1+0 = 0+1 = 1 George Boole • A powerful tool that can be used for • 4a: If x=0, then x’ = 1 designing and analyzing logic circuits 1815-1864 4b: If x=1, then x’ = 0 19 20 Single-Variable Theorems Two- and Three-Variable Properties • 5a: x.0 = 0 Null • 10a: x.y = y.x Commutative 5b: x+1 = 1 10b: x+y = y+x • 6a: x.1 = x Identity • 11a: x.(y.z) = (x.y).z Associative 6b: x+0 = x 11b: x+(y+z) = (x+y)+z • 7a: x.