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PROGRAM TITLE: Btec-Computing UNIT TITLE: Unit 18 - Discrete Maths ASSIGNMENT NUMBER: 03 ASSIGNMENT NAME: ASSIGNMENT 3- Boolean algebra. SUBMISSION DATE: 4/11/2021. TUTORIAL LECTURER: Lưu Thị Hương Giang. WORD COUNT: 1611. STUDENT NAME: Nguyễn Bích Hạnh STUDENT ID: BKC 1985
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A. CONTENT:
Part 1: Discuss two real world binary problems in two different fields using applications of Boolean algebra.
- Definition: In mathematical logic, Boolean algebra is a branch of algebra in which the values of the variables are the truth values True(T) and False(F), usually denoted 1 and 0 respectively. basic, where the values of variables are numbers and the prime operations are addition and multiplication, the main operations of Boolean algebra are association (and) denoted ∧, division (or) is denoted by ∨ and the negation (zero) is denoted by ¬. It is therefore a form of describing logical operations, in the same way that elementary algebra describes numerical operations.
- Discuss two real world binary problems in two different fields using applications of Boolean algebra:
- The simplest Boolean algebra is that defined on the set {0,1} and 0 signifies OFF and 1 signifies ON states. Each segment of a calculator's display is switched ON and OFF by a series of logic gates that are connected together. Consider just the bottom lower right segment need to turn this segment on if we're showing the numbers 0 (binary 00), 1 (01), 3 (11), 4 (100), 5 (101), 6 (110), 7 (111), 8 (1000), and 9 (1001)
- But not if show the number 2 (10). Can make the segment switch on and off correctly for the numbers 1– 10 by rigging up OR gates and one NOT gate like this. a. Representing numbers in binary:
- For the last century or so, computers and calculators have been built from a variety of switching devices that can either be in one position or another. It has only “one (1)” or “zero (0)”. For that reason, computers and calculators store and process numbers using what's calledbinary code, which uses just two symbols ( and 1) to represent any number.
- The most common computer architectures use ordered sequences of Boolean values, called bits, of 32 or 64 values, for example: 01101000110101100101010101001011. When programming in machine code, assembly language, and a number. In other programming languages, programmers work with the low-level numerical structure of data registers. These registers operate on voltages, where volt 0 represents Boolean 0 and the reference voltage (usually +5 V, +3.3 V, +1.8 V) represents Boolean 1. Such languages Supports both numeric and logical operations. In this context, "number" means that the computer treats strings of bits as binary numbers (base two) and performs arithmetic operations such as addition, subtraction, multiplication, or division. "Logical" refers to the Boolean logical operations of the separation, union, and negation between 2 bit strings, where each bit in one string is simply compared with its counterpart in the other. Thus, programmers have the option of working and applying the rules of algebra or Boolean algebra
as needed. A key distinguishing feature between these families of activities is the existence of an operation that performs in the first but not the second. So in binary code, the number 56 is written is 111000, Which means: 56 / 2 = 28 with 0 remainder 28 / 2 = 14 with 0 remainder 14 / 2 = 7 with 0 remainder 7 / 2 = 3 with 1 remainder 3 / 2 = 1 with 1 remainder 1 / 2 = 0 with 1 remainder b. Using logic gates with binary: When you want to do the sum 3 + 2 = 5. A calculator tackles a problem like this by turning the two numbers into binary, giving 11 (which is 3 in binary = 1 × 2 + 1 × 1) plus 10 (2 in binary = 1 × 2 + 0 × 1) makes 101 (5 in binary = 1 × 4 + 0 × 2 + 1 × 1).So how does the calculator do the actual sum, it uses logic gates to compare the pattern of switches that are active and come up with a new pattern of switches instead. A logic gate is really just a simple electrical circuit that compares two numbers and produces a third number depending on the values of the original numbers.
- Logic gates: Switching circuits form the basis of computer hardware. Usually, a high voltage represents TRUE (or 1) while a low voltage represents a FALSE (or 0). Digital circuits can be represented using letters for each input. a. The three basic gates: There are three basic gates which combine inputs and represent the operators NOT(−), AND (.), and OR (+).
- An OR gate has two inputs and it produces an output of 1 if either of the inputs is 1; it produces a zero otherwise.
- An AND gate also has two inputs, but it produces an output of 1 only if both inputs are 1.
- An NOT gate has a single input and reverses it to make an output. So if feed it a zero, it produces a 1.
W: " warning_light " =>W = D. 𝐵̅ .I D B I (^) 𝐵̅ W = D. 𝐵̅ .I 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 0 0 b) If it rains and you don't open your umbrella then you will get wet. We have:s R: "rains" U: "open_umbrella" W: "Wet" =>W =R. 𝑈̅ R U (^) 𝑈̅ W = R. 𝑈̅ 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 0 2.Produce truth tables for given Boolean expressions. i. 𝐴̅ 𝐵̅ C+AB𝐶̅ + ABC +𝐴̅ B𝐶̅
A B C 𝐴̅ 𝐵̅ 𝐶̅ 𝐴̅ 𝐵̅ C AB𝐶̅ ABC 𝐴̅ B𝐶̅ 𝐴̅ 𝐵̅ C+AB𝐶̅ +ABC+𝐴̅ B𝐶̅
ii. (A+𝐵̅ +C) (A+B+C) ( 𝐴̅ + B + 𝐶̅ ) A B C 𝐴̅ 𝐵̅ 𝐶̅ A+𝐵̅ +C A+B+C 𝐴̅ + B + 𝐶̅ (A+𝐵̅ +C) (A+B+C) ( 𝐴̅ +B + 𝐶̅ ) 0 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 Part 3: Find the simplest form of given Boolean expressions using algebraic methods. i. A(A+B) +B(B+C) +C(C+A) = A.A + A.B +B.B +B.C+C.C+C.A = A +AB+B+BC+C+CA [We have: A.A =A; B.B =B; C.C =C (Idempotent Law)] = A(1+B) +B(1+C) +C(1+A) [We have: 1+B = 1; 1+C=1; 1+A=1(Bound Laws)] =A+B+C ii. (A+𝐵̅ )(B+C) + (A+B) (C+𝐴̅ ) =AB+AC+𝐵̅ B+𝐵̅ C+AC+A𝐴̅ +BC+B𝐴̅ [We have: 𝐵̅ B =0; A𝐴̅ =0(Complement laws)] =AB+AC+𝐵̅ C+AC+BC+B𝐴̅ [We have: AC+AC = AC (Idempotent Law)]
