10/22/2004EE 42 fall 2004 lecture 221 Lecture #22 Truth tables and gates This week: Circuits for digital devices

10/22/2004EE 42 fall 2004 lecture 222 Topics Today: Combinatorial logic Truth tables And, Or, and Not gates

10/22/2004EE 42 fall 2004 lecture 223 Combinatorial logic Combinatorial logic describes a digital circuit where there are a set of digital inputs, say N wires each one (1) or zero (0) ( example:1=true, 2 volts or 0=false, zero volts) And M digital outputs, M wires each carrying 1 or 0, which are an instantaneous function of the inputs So in combinatorial logic, there is no sequence of events, internal memory or flags, just a straight input  output

10/22/2004EE 42 fall 2004 lecture 224 Truth Table A truth table is a general way of describing combinatorial logic, by just listing all of the possible states of the input, and the value of each output which is the result. Lets look at a truth table for “exclusive or” (XOR) ABoutput Every possible combination of inputs

10/22/2004EE 42 fall 2004 lecture 225 In principle, every problem which could be described with discrete inputs (integers, fractions, flags programs!) could be solved with a single combinatorial logic machine. This is very fast once built. But it practice, the combinatorial logic would get too complex, for example the truth table would have 2 n rows, where N is the number of Boolean variables needed to take into account all possible inputs. So for complex problems, we use combinatorial logic circuits as steps from state to state of a machine (a “finite state machine”, for example a computer)

10/22/2004EE 42 fall 2004 lecture 226 Translating mathematics to machines So once again, we are in the position of translating mathematics into a machine which can execute the formulas, but this time as digital, Boolean expressions rather than as continuous functions of time and voltage.

10/22/2004EE 42 fall 2004 lecture 227 Logical expressions Fortunately, as Boole pointed out, the language of facts which are true or false are natural to us as a species, and so we can deal with much of Boolean logic intuitively. However, more complex logic expressions are easier if we have a notation, symbols and rules for manipulation.

10/22/2004EE 42 fall 2004 lecture 228 Truth tables with 2 inputs With just two Boolean inputs, there are four possible combinations, so a truth table for two inputs would have four rows. 00, 01, 10, 11 Each of the rows of a possible truth table can have a different Boolean output, so there are 16 different possible truth tables for an expression with two inputs, and they are shown on the next slide

9 ABout AB AB AB AB AB AB AB AB AB AB AB AB AB AB AB NOR AND XOR OR NAND

10/22/2004EE 42 fall 2004 lecture 2210 And, Or, and Not are sufficient As you see from the next slide, with only the functions AND OR and NOT, all of the possible expressions for two inputs can be formed. Any expression of any number of inputs can be formed using just AND OR, and NOT. NOR by itself is also complete, but is not as intuitive to use

11 ABout AB AB AB AB AB AB AB AB AB AB AB AB AB AB AB Not (A or B) AND (A or B) and (not (A and B)) OR Not (A and B) FalseB and (not A)Not A A and (not B)Not B Not (A OR B)B(not A) or (A and B) AA OR (Not B)True

10/22/2004EE 42 fall 2004 lecture 2212 NOR is sufficient by itself NOT A = A NOR A A AND B = (Not A) NOR (Not B) A OR B= Not (A NOR B) So if you can build a NOR circuit, these can be combined to form any Boolean logic expression

10/22/2004EE 42 fall 2004 lecture 2213 Logical Expressions Example: Z = A Examples: X = A · B ; Y = A · B · C Examples: W = A+B ; Z = A+B+C Standard logic notation : AND:“dot” OR :“+ sign” NOT:“bar over symbol for complement” With these basic operations we can construct any logical expression. Order of operation: NOT, AND, OR (note that negation of an expression is performed after the expression is evaluated, so there is an implied parenthesis, e.g. means.

10/22/2004EE 42 fall 2004 lecture 2214 Logic Function Example Boolean Expression:H = (A · B · C) + T This can be read H=1 if (A and B and C are 1) or T is 1, or H is true if all of A,B,and C are true, or T is true, or The voltage at node H will be high if the input voltages at nodes A, B and C are high or the input voltage at node T is high

10/22/2004EE 42 fall 2004 lecture 2215 Logic Function Example 2 Boolean Expression:B = A + S(D + T ) This can be read B=1 if A = 1 or S=1 AND (D OR T =1), i.e. B=1 if {A = 1} or {S=1 AND (D OR T =1)} or B is true IF {A is true} OR {S is true AND D OR T is true} or The voltage at node H will be high if {the input voltage at node A is high} OR {the input voltage at S is high and the voltages at D and T are high} You wish to express under which conditions your burglar alarm goes off (B=1): If the “Alarm Test” button is pressed (A=1) OR if the Alarm is Set (S=1) AND { the door is opened (D=1) OR the trunk is opened (T=1)}

10/22/2004EE 42 fall 2004 lecture 2216 Example “Truth Table” Truth Table for Logic Expression H = (A · B · C) + T

10/22/2004EE 42 fall 2004 lecture 2217 Evaluation of Logical Expressions with “Truth Tables” The Truth Table completely describes a logic expression The Truth Table is the fundamental meaning of a logic expression. Two logic expressions are equal if their truth tables are the same

10/22/2004EE 42 fall 2004 lecture 2218 Some Important Logical Functions “AND” “OR” “INVERT” or “NOT” “not AND” = NAND “not OR” = NOR exclusive OR = XOR (or ) AnotA

10/22/2004EE 42 fall 2004 lecture 2219 These are circuits that accomplish a given logic function such as “OR”. We will shortly see how such circuits are constructed. Each of the basic logic gates has a unique symbol, and there are several additional logic gates that are regarded as important enough to have their own symbol. The set is: AND, OR, NOT, NAND, NOR, and EXCLUSIVE OR. Logic Gates A B C=A·B AND C = A B NAND C = NOR A B NOT A OR A B C=A+B EXCLUSIVE OR A B

10/22/2004EE 42 fall 2004 lecture 2220 With a combination of logic gates we can construct any logic function. In these two examples we will find the truth table for the circuit. Logic Circuits B A C It is helpful to list the intermediate logic values (at the input to the OR gate). Let’s call them X and Y. X Y Now we complete the truth tables for X and Y, and from that for C. (Note that X and Y and finally C = X + Y) ABXYC Interestingly, this is the same truth table as the EXCLUSIVE OR