NOT AND OR XOR NAND NOR

Transcription

1 NOT AND OR XOR NAND NOR

2 Expression 1: It is raining today Expression 2: Today is my birthday X Meaning True False It is raining today It is not raining Binary representation of the above: X Meaning 1 It is raining today 0 It is not raining

3 Truth Tables: A table that shows the results of applying the logical function to all possible combinations of inputs. X Y Q Boolean equation: An equation that expresses a Boolean output Q in terms of Boolean inputs X, Y, Z etc. To which one or more Boolean functions such as OR, AND, and NOT are applied. For example X+Y=Q

4 =0 Supply Lamp Supply Lamp =1

5 OR Function Supply X Lamp Q Truth Table Y X Y Q

6 OR Function Supply X Lamp Q Black-box Diagram Y X Y OR Q

7 AND Function Supply X Y Q Lamp Black-box Diagram X AND Q

8 AND Function Supply Truth Table X Y Q Lamp X Y Q

9 NOT Function Truth Table X Q Q= NOT X or Q = X X Y Q

10 NOT Function Black-Box Diagram X NOT Q

11 Combination of AND and OR Supply X Y Lamp Q Boolean equation from the above arrangements: Z Q=(X AND Y) OR (X AND Z) Q=X.Y + X.Z Q= x.(y+z)

12 1. Draw the truth tables for the AND, OR and NOT functions 2. What logic function is performed by the switch arrangement shown below? Supply X Y Lamp Q 3. Draw a switch arrangement for the AND function 4. Draw the switch arrangement for output Q where Q=X.Y + X.Z

13 Logic Gate is an electronic circuit that performs the Boolean function. Symbols used: Q=A.B Q=A A B Q A Q AND Gate NOT Gate A B Q A B Q A B Q Q=A+B OR Gate NOR Gate Q=A+B NAND Gate Q=A.B

14 A B C D E F Q Q=(A.B)+(C.D) Q=E+F

15 A B C D E F Q Suppose we require an output Q from three inputs A, B, C such that Q= 0 when A is present (1) and B is present (1) OR When B is not present (0) but C is present (1) Q= A.B + B.C

16 A B A.B A.B+ B. C=Q Q B B.C C Q= A.B+B.C

17 A B Q=A B Also known as EX-OR or XOR function Differs slightly from the normal OR function, often called the inclusive OR function The inclusive OR function gives an output of 1 if any input is 1 The exclusive OR function gives an output of 1 if only one of the inputs is 1 and not if both inputs are 1 Input A Input B Output XOR Function Input A Input B Output OR Function

18 A gate which performs the XOR function is called an exclusive OR, EX-OR or XOR gate The XOR gate is sometimes known as a non-equivalence gate as its output is 1 if the inputs are not equivalent Boolean algebra representation of the exclusive OR function is: Q = A.B + A.B Condition: A=1, B=0 A.B i.e. A AND [NOT B] Condition: A=0, B=1 A.B i.e. [NOT A] AND B therefore Q=1

19 A B A.B Boolean function, which is simply a combination of the AND and NOT functions NOT AND abbreviated to NAND A.B Indicates inverted output NAND function: logic circuit A B A.B A.B Truth Table Input A Input B Output

20 A B A+B Indicates inverted output NOT OR function NOR function: logic circuit A B A+B A+B Truth Table Input A Input B Output

21 1. Draw truth tables for the NAND, NOR and XOR logic gates 2. Draw the circuit symbols for the AND, OR, NOT, NAND NOR and XOR logic gates 3. EXPRESS q IN TERMS OF Boolean variables A, B and C for the arrangements of logic gates below: A B Q C

22 1. Construct a logic circuit using only NAND gates for the Boolean expression (A.B).(B.A) 2. Construct logic circuits, using only NOR gates, for the Boolean expression (A+B)+(A+B)

23 Enable Boolean expressions to be converted to forms requiring only the OR and NOT functions or only the AND and NOT functions This means any Boolean expression may be implemented using only OR gates and NOT gates [NOR gates] or using only AND gates and NOT gates [NAND gates] NAND and NOR gates may be used as NOT gates by connecting all the gate inputs together

24 Therefore, NOR gates alone or NAND gates alone can implement any Boolean function The fabrication on a single chip of many NOR gates or many NAND gates is possible with integrated circuit technology

25 In general, De Morgan s Laws may be written as follows: A 1 + A 2 + A A n = A 1.A 2.A 3..A n A 1.A 2.A 3..A n = A 1 +A 2 +A 3 + +A n In simplest form : A + B = A.B A.B = A + B

26 The laws are easily remembered as everything associated with an expression must be changed: The logical connectives must be changed: AND to OR and OR to AND The logical state of each variable must be changed: A to A, A to A The logical state of the complete expression must be changed: expression to expression, expression to expression

27 1. Show that A.B+A.B = (A+B).(A+B), where A and B are Boolean variables 2. Show that (A+B).(A+B) = (A+B).(A+B) by using A.A = B.B = 0 3. Simplify 1.B where B is a Boolean variable and 1 is a Boolean constant 4. Show that (A.A.B).(B.A.B) = A.B + B.A 5. Output Q = A.(C+D)+B.(C+D) where A, B, C and D are Boolean inputs. Show how it is possible to use just four NAND gates to produce output Q from A, B, C and D. One of the NAND gates is used as a NOT gate

28 In this topic we have covered: Boolean variables may have two discrete possible values, e.g. true or false, 1 or 0 A truth table shows the result of applying a logical function to all possible input combinations OR function (+): output is true if either or both inputs are true AND function (.): the output is true if all inputs are true NOT function ( ): the output is the inversion of the input XOR function( ): the output is true if either input is true but not if both are true

29 In this topic we have covered: NAND function (NOT AND): the output is true if any input is false NOR function (NOT OR): the output is true only when all inputs are false Logic gates perform Boolean functions such as AND, OR and NOT Output Q is expressed in terms of Boolean inputs X, Y, Z etc. and Boolean functions De Morgan s Laws: A+B = A.B and A.B=A+B To simplify a logical expression use a truth table to write down input combinations that make the output 1 then use De Morgan s Law to produce a solution in NAND or NOR

30 In this topic we have covered: The logic gate symbols: