Boolean Functions (10.1) Representing Boolean Functions (10.2) Logic Gates (10.3)

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1 Chapter (Part ): Boolean Algebra Boolean Functions (.) Representing Boolean Functions (.2) Logic Gates (.3)

2 It has started from the book titled The laws of thought written b George Boole in 854 Claude Shannon showed how the basic rules of logic could be used to design circuits The circuit i in electronic devices have inputs, each of which is either or and produce outputs that are also or

3 The operation of a circuit is defined b a Boolean function that specifies the value of an output for each set of inputs One of the goal is to describe methods for finding a simplified epression (min number of sums and products) that represents a Boolean function (Karnaugh maps).

4 Boolean Functions (.) Introduction Electronic and optimal switches are studied using the set {, } and the rules of Boolean algebra The 3 operations in Boolean algebra that are used are Complementation, Boolean Sum and the Boolean Product

5 Boolean Functions (.) (cont.) Complement of an element is defined b: = ; = Boolean sum denoted b or b OR obes: =, =, =, = Boolean product denoted b AND obes:. =,. =,. =,. =.

6 Boolean Functions (.) (cont.) Eample: Find the value ( ) Solution: ( ) = = = Boolean epressions & Boolean Functions Definition Let B = {, }thenb n = {(, 2,, n ) such that i B Let B {, } then B {(, 2,, n ) such that i B for i n}. The variable is called a Boolean variable if it assumes values onl from B ( or ). A function B n to B is called a Boolean function of degree n.

7 Boolean Functions (.) (cont.) Eample:The function F (, ) = from B 2 ={(, ) such that t (, ) {, }*{ {,)} is a Boolean function with: F(, ) =, F(, ) =, F(, ) =, F(, ) = Eample:Find the values of the Boolean function represented b (,, ) = F Solution: The values of this function are displaed on the following table:

8 Boolean Functions (.) (cont.) ),, ( F = ),, ( F

9 Identities of Boolean algebra Identit Name = =. = =. = =. = = = ( ) = ( ) () = () = ( ) ( ) Law of the double component Idempotent laws Identit laws Domination laws Commutative laws Associative laws Distributive laws ( ) = ( ) = De Morgan slaws = ( ) = Absorption laws = = Unit propert Zero propert

10 Boolean Functions (.) (cont.) Eample: Show that the distributive law ( ) = Solution: ( ) ( )

11 Dualit Boolean Functions (.) (cont.) Goal: the dual of a Boolean epression is obtained b interchanging Boolean sums and Boolean products and interchanging s and s Eample: Find the duals of ( ) and. ( ) Solution: the duals are (.) and ( )( )

12 Boolean Functions (.) )(cont.) The dualit principlep enables to obtain more identities. Since an identit between functions represented b Boolean epressions remains valid when the duals of both sides of the identit i are taken The abstract definition of a Boolean Algebra Goal: Recognie that a particular structure (set, propositions, ) is a Boolean algebra all results of Boolean algebra will appl

13 Boolean Functions (.) (cont.) Dfiii Definition : A Boolean algebra is a set B with two binar operations and, elements and, and a unar operation such that these properties p hold for all, and in B: o = = ( ( Identit laws = Complement laws = ) = ) = = = ( ) = ( ( ) = ( ( ) Associative laws ( ) Commutative laws ) ( ) Distributiveib i laws ) ( )

14 Boolean Functions (.) (cont.) The set of propositions p in n variables, with the and operators, F and T, and the negation operator, also satisfies all the properties p of a Boolean algebra Similarl, the set of subsets of a universal set Ω with the union and intersection operators, the empt set and the universal set, and the set complementation operator is a Boolean algebra

15 Representing Boolean Functions (.2) Two problems of Boolean algebra are emphasied in this section. Given the values of a Boolean function, how can a Boolean epression that represents this function be found? 2 Is there a smaller set of operators that can be 2. Is there a smaller set of operators that can be used to represent all Boolean functions?

16 Representing Boolean Functions (.2) (cont.) Sum-of-products epansions G l Fi d B l i th t t Goal: Find a Boolean epression that represents a Boolean function Eample:Find Boolean epressions that represent the functions F(,, ) and G(,, ) p (,, ) (,, ) given b the following table: F G

17 F G Representing Boolean Functions (.2) (cont.) Solution: G To form the Boolean epressions for the function F(,,), we need to look at the table and notice that F has the value onl if = and = and = otherwise F has value F(,,) = To form the Boolean epression for the function G(,,), we have to notice that G(,,) = = = and = or = = and = ; otherwise G(,,) = G()= G(,,) ( ) ( )

18 Representing Boolean Functions (.2) (cont.) Definition : A literal is a Boolean variable or its complement. A minterm of the Boolean variables =, 2,, n is a Boolean product 2 n, where i = i or i = i. Hence, a minterm is a product of n literals, with one literal l of each variable. The minterm 2 n, is if and onl if each i is, and therefore i = when i = i and i = when i = i.

19 Representing Boolean Functions (.2) (cont.) Eample: Find a minterm that equals if = 3 = and 2 = 4 = 5 =, and equals otherwise Solution: The minterm is: Minterms are used to determine the Boolean epression of a Boolean function in a table

20 Representing Boolean Functions (.2) (cont.) Eample: Find the sum-of-products epansion for the function F (,, ) = ( ) Solution (): ( ) = = = ( ) ( ) = = (since u u = u)

21 Representing Boolean Functions (.2) (cont.) Solution (2): We use the minterm technique from atable a table ( )

22 Representing Boolean Functions (.2) (cont.) We need onl to spot the set of values (,,) for which This set is: F (,, ) = ( ) =. {(,,);(,,);(,,)} We need to find the minterm that equals if ( = = and = )OR ( = and = = )OR ( = = and = ) Theorem : Ever Boolean function can be represented using the three Boolean operators., and, (complement)

23 Representing Boolean Functions (.2) (cont.) Functional completeness The set{.,, - } is functionall complete because of theorem Question: Can we find a smaller set of functionall complete operators? We answer es to this question if one of the three operators can be epressed in terms of the other two One of the De Morgan s laws provide the solution: = since ( ) =

24 Representing Boolean Functions (.2) (cont.) Similarl, we have: = Finall, we have found a smaller set functionall complete; it contains onl 2 operators! Question: Can we still reduce the set to onl one operator? The answer is es. Indeed, there are 2 sets that contain onl one operator, the are: the NAND and the NOR.

25 Representing Boolean Functions (.2) (cont.) NAND ( ) is defined as = and = = = NOR ( ) is defined as = = = and = Since the set {., - } is functionall complete and =, = ( ) ( ) then the set { } is functionall complete. Home-eercise: Prove that the set { } is functionall complete.

26 Introduction Logic Gates (.3) A computer, or electronic devices, is made up of a number of circuits Each circuit can be designed using the rules of Boolean algebra The basic element of circuits are called gates Each tpe of gates implements a Boolean Each tpe of gates implements a Boolean Operation

27 Logic Gates (.3) (cont.) All circuits studied in this chapter provides output that depends onl on the input, and not on the current state of the circuit These circuits that have no memor capabilities are called combinational circuits or gating networks Combinatorial i circuits i are built using 3 tpes of elements: () inverter, (2) OR gate, and d(3) AND gate

28 Logic Gates (.3) (cont.) Role of each tpe of elements:. Inverter: it accepts the value of one Boolean variable as input and produces the complement of this value as its output.

29 Logic Gates (.3) (cont.) 2. OR gate: it accepts as input values of 2 or more Boolean variables. The output is the Boolean sum of their values

30 Logic Gates (.3) (cont.) 3. AND gate: The inputs to this gate are the values of two or more Boolean variables. The output is the Boolean product of their values.

31 Logic Gates (.3) (cont.) We can have multiple inputs to AND and OR: 2 n 2 n 2 n 2 n Gates with n inputs

32 Logic Gates (.3) (cont.) Combinations of Gates Combinational circuits can be constructed using a combination i of inverters, OR gates, and AND gates When combinations of circuits are formed, some gates ma share input using branching The following figure shows 2 was of drawing the same circuit

33 Two was to draw the same circuit

34 Logic Gates (.3) (cont.) Eample:Construct circuits that produce the following outputs: a) ( ) b) ( ) c) ( )( )

35 Logic Gates (.3) (cont.) Solution for (a): ( )

36 Logic Gates (.3) (cont.) Solution for (b): ( ) ( )

37 Logic Gates (.3) (cont.) Solution for (c): ( )( )

38 Logic Gates (.3) (cont.) Eample of circuits Eample:A committee of 3 individuals decides issues for an organiation. Each individual votes either es or no for each proposal that arises. A proposal p is passed if it receives at least 2 es votes. Design a circuit that determines whether a proposal p passes. Solution: We want to have the following Boolean function: represented b the following circuit:

39 Logic Gates (.3) (cont.) A circuit for majorit voting

24 Nov Boolean Operations. Boolean Algebra. Boolean Functions and Expressions. Boolean Functions and Expressions

24 Nov Boolean Operations. Boolean Algebra. Boolean Functions and Expressions. Boolean Functions and Expressions 24 Nov 25 Boolean Algebra Boolean algebra provides the operations and the rules for working with the set {, }. These are the rules that underlie electronic circuits, and the methods we will discuss are