al ict m3 logic first

7/30/2019 AL ICT m3 Logic First

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Mathematics for Computing

Faculty of Information Technology - University of Moratuwa 73

'83' c 'A3' t 'D0' } 'E9' Z

'84' d 'A4' u 'D1' J 'F0' 0

'85' e 'A5' v 'D2' K 'F1' 1

'86' f 'A6' w 'D3' L 'F2' 2

'87' g 'A7' x 'D4' M 'F3' 3

'88' h 'A8' y 'D5' N 'F4' 4

'89' i 'A9' z 'D6' O 'F5' 5

'91' j 'C0' { 'D7' P 'F6' 6

'92' k 'C1' A 'D8' Q 'F7' 7

'93' l 'C2' B 'D9' R 'F8' 8

'94' m 'C3' C 'E2' S 'F9' 9

Table 2.9-1

2.10 Boolean Algebra

The most obvious way to simplify Boolean expressions is to manipulate them in thesame way as normal algebraic expressions are manipulated. With regards to logicrelations in digital forms, a set of rules for symbolic manipulation is needed in order to solve for the unknowns.A set of rules formulated by the English mathematician George Boole describe certain

propositions whose outcome would be either true or false . With regard to digitallogic, these rules are used to describe circuits whose state can be either, 1 (true) or 0(false) . In order to fully understand this, the relation between the AND gate, OR gateand NOT gate operations should be appreciated.

OR operation:

The OR operation says if any input is on, the output will be on. It's easy to see all of the combinations by using what are called truth tables, illustrated below.

Input 1 Input 2 Output0 0 00 1 11 0 11 1 1

AND operation:


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The AND operation says if and only if all inputs are on, the output will be on. Theoutput will be off if any of the inputs are off.

Input 1 Input 2 Output0 0 0

0 1 01 0 01 1 1

NOT operation:

The inverter or NOT operation, says that the output will be opposite in state to theinput. It obviously has only one input and one output. Note that it will change anAND to a NAND, an OR to a NOR and an XOR to a NXOR if connected to their outputs. It simply changes 1s to 0s and 0s to 1s.

Input 1 Output0 11 0

2.11 Basic Laws and Axioms of Boolean algebra

Note that every law has two expressions, (a) and (b). This is known as duality . Theseare obtained by changing every AND (.) to OR (+), every OR (+) to AND (.) and all1's to 0's and vice-versa. It has become conventional to drop the. (AND symbol) i.e.

A.B is written as AB.

T1 : Commutative Law(a) A + B = B + A(b) A B = B A

T2 : Associate Law(a) (A + B) + C = A + (B + C)(b) (A B) C = A (B C)

T3 : Distributive Law(a) A (B + C) = A B + A C (b) A + (B C) = (A + B) (A + C)

T4 : Identity Law(a) A + A = A(b) A A = A

T5 :(a)(b)

T6 : Redundancy Law


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(a) A + A B = A(b) A (A + B) = A

T7 :(a) 0 + A = A(b) 0 A = 0

T8 :(a) 1 + A = 1(b) 1 A = A

T9 :

(a)(b)

T10 :(a)

(b)

T11 : De Morgan's Theorem

(a)

(b)

2.12 Truth Tables

A truth table shows how a logic circuit's output responds to various combinations of the inputs, using logic 1 for true and logic 0 for false. All permutations of the inputsare listed on the left, and the output of the circuit is listed on the right. The desiredoutput can be achieved by a combination of logic gates. A truth table for two inputs isshown, but it can be extended to any number of inputs. The input columns are usuallyconstructed in the order of binary counting with a number of bits equal to the number of inputs.

Truth Table for AND Gate

X Y Output0 0 00 1 01 0 01 1 1

AND OR NOT Operations

.. (Insert from previous document)

Boolean Expressions

(Insert from previous document)


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2.13 Logic Gates and Circuits

Gates and simple circuitsThe term Gate is used to describe the members of a set of basic electroniccomponents which, when combined with each other, are able to perform complexlogical and arithmetic operations. 'Gates' are the physical realization of the simpleBoolean expressions. Electronic circuits which combine digital signals according tothe Boolean algebra are referred to as logic gates ; gates because they control the flowof information.

Example logic circuits

Gate Symbol Truth Table


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Design logic circuit for Boolean expression F CD B A =+

In this example A,B,C,D are inputs and F is out put

B Can be represent with

B A expression having AND logic, we can take P B A =

CD expression also having AND logic, we can take QCD =

Then final expression would be in form of P+Q = F, This has OR logic

Therefore we can design final circuit for the Boolean expression F CD B A =+ as

al ict m3 logic first

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