SISTEM BILANGAN

SISTEM BILANGAN

I. DEFINISISystem bilangan (number system) adalah suatu cara untuk mewakili besaran dari suatu item fisik. Sistem bilanan yang banyak dipergunakan oleh manusia adalah system biilangan desimal, yaitu sisitem bilangan yang menggunakan 10 macam symbol untuk mewakili suatu besaran.Sistem ini banyak digunakan karena manusia mempunyai sepuluh jari untuk dapat membantu perhitungan. Lain halnya dengan komputer, logika di komputer diwakili oleh bentuk elemen dua keadaan yaitu off (tidak ada arus) dan on (ada arus). Konsep inilah yang dipakai dalam sistem bilangan binary yang mempunyai dua macam nilai untuk mewakili suatu besaran nilai.Selain system bilangan biner, komputer juga menggunakan system bilangan octal dan hexadesimal. II. Teori Bilangan

1. Bilangan DesimalSistem ini menggunakan 10 macam symbol yaitu 0,1,2,3,4,5,6,7,8,dan 9. system ini menggunakan basis 10. Bentuk nilai ini dapat berupa integer desimal atau pecahan. Integer desimal :adalah nilai desimal yang bulat, misalnya 8598 dapat diartikan :8 x 103 = 80005 x 102= 5009 x 101= 908 x 100= 8 8598position value/palce valueabsolute value

Absolue value merupakan nilai untuk masing-masing digit bilangan, sedangkan position value adalah merupakan penimbang atau bobot dari masing-masing digit tergantung dari letak posisinya, yaitu nernilai basis dipangkatkan dengan urutan posisinya.

Pecahan desimal :Adalah nilai desimal yang mengandung nilai pecahan dibelakang koma, misalnya nilai 183,75 adalah pecahan desimal yang dapat diartikan :1 x 10 2= 1008 x 10 1= 803 x 10 0= 37 x 10 1= 0,75 x 10 2= 0,05 183,75

2. Bilangan BinarSistem bilangan binary menggunakan 2 macam symbol bilangan berbasis 2digit angka, yaitu 0 dan 1. Contoh bilangan 1001 dapat diartikan :

1 0 0 11 x 2 0= 10 x 2 1= 00 x 2 2= 01 x 2 3= 810 (10)Operasi aritmetika pada bilangan Biner :a. Penjumlahan Dasar penujmlahan biner adalah :0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 0dengan carry of 1, yaitu 1 + 1 = 2, karena digit terbesar ninari 1, maka harus dikurangi dengan 2 (basis), jadi 2 2 = 0 dengan carry of 1contoh :

1111 10100 + 100011atau dengan langkah : 1 + 0= 1 1 + 0= 1

1 + 1= 0 dengan carry of 1 1 + 1 + 1= 0

1 + 1= 0 dengan carry of 1 1 0 0 0 1 1

b. PenguranganBilangan biner dikurangkan dengan cara yang sama dengan pengurangan bilangan desimal. Dasar pengurangan untuk masing-masing digit bilangan biner adalah :0 - 0 = 01 - 0 = 11 - 1 = 00 1 = 1dengan borrow of 1, (pijam 1 dari posisi sebelah kirinya).Contoh :11101 1011 - 10010

dengan langkah langkah :1 1= 0

0 1= 1 dengan borrow of 11 0 1= 01 1= 01 0= 11 0 0 1 0

c. PerkalianDilakukan sama dengan cara perkalian pada bilangan desimal. Dasar perkalian bilangan biner adalah :0 x 0 = 01 x 0 = 00 x 1 = 01 x 1 = 1contohDesimalBiner

14 12 x 28 14

+168

1110 1100 x 0000 0000 1110 1110 + 10101000

d. pembagianPembagian biner dilakukan juga dengan cara yang sama dengan bilangan desimal. Pembagian biner 0 tidak mempunyai arti, sehingga dasar pemagian biner adalah :0 : 1 = 01 : 1 = 1DesimalBiner

5 / 125 \ 25 10 - 25 25 - 0 101 / 1111101 \ 11001 101 - 101 101 - 0101 101 - 0

3. Bilangan OktalSistem bilangan Oktal menggunakan 8 macam symbol bilangan berbasis 8 digit angka, yaitu 0 ,1,2,3,4,5,6,7. Position value system bilangan octal adalah perpangkatan dari nilai 8.Contoh :12(8) = (10)2 x 8 0 = 21 x 8 1=810Jadi 10 (10)

Operasi Aritmetika pada Bilangan Oktala. PenjumlahanLangkah-langkah penjumlahan octal : tambahkan masing-masing kolom secara desimal rubah dari hasil desimal ke octal tuliskan hasil dari digit paling kanan dari hasil octal kalau hasil penjumlahan tiap-tiap kolom terdiri dari dua digit, maka digit paling kiri merupakan carry of untuk penjumlahan kolom selanjutnya.Contoh :DesimalOktal

21 87 + 108 25127 + 154 5 10 + 7 10 = 12 10 = 14 8 2 10 + 2 10 + 1 10 = 5 10 = 5 8 1 10 = 1 10 = 1 8

b. PenguranganPengurangan Oktal dapat dilaukan secara sama dengan pengurangan bilangan desimal.Contoh :DesimalOktal

108 87 - 21 154127 - 25 4 8 - 7 8 + 8 8 (borrow of) = 5 8 5 8 - 2 8 - 1 8 = 2 8 1 8 - 1 8 = 0 8

c. PerkalianLangkah langkah : kalikan masing-masing kolom secara desimal rubah dari hasil desimal ke octal tuliskan hasil dari digit paling kanan dari hasil octal kalau hasil perkalian tiap kolol terdiri dari 2 digit, maka digit paling kiri merupakan carry of untuk ditambahkan pada hasil perkalian kolom selanjutnya.Contoh :DesimalOktal

14 12 x 28 14 + 168

16 14 x 70 4 10 x 6 10 = 24 10 = 30 8 4 10 x 1 10 + 3 10 = 7 10 = 7 8

16 14 x 70 16 1 10 x 6 10 = 6 10 = 6 8 1 10 x 1 10 = 1 10 = 1 8

16 14 x 70 16 + 250 7 10 + 6 10 = 13 10 = 15 8 1 10 + 1 10 = 2 10 = 2 8

d. PembagianDesimalOktal

12 / 168 \ 1412 - 48 48 0 14 / 250 \ 16 14 - 14 8 x 1 8 = 14 8 110 110 - 14 8 x 6 8 = 4 8 x 6 8 = 30 8 0 1 8 x 6 8 = 6 8 + 110 8

4. Bilangan HexadesimalSistem bilangan Oktal menggunakan 16 macam symbol bilangan berbasis 8 digit angka, yaitu 0 ,1,2,3,4,5,6,7,8,9,A,B,C,D,Edan FDimana A = 10, B = 11, C= 12, D = 13 , E = 14 dan F = 15Position value system bilangan octal adalah perpangkatan dari nilai 16.Contoh :C7(16) = (10)7 x 16 0 = 7C x 16 1= 192 199Jadi 199 (10)

Operasi Aritmetika Pada Bilangan Hexadesimala. PenjumlahanPenjumlahan bilangan hexadesimal dapat dilakukan secara sama dengan penjumlahan bilangan octal, dengan langkah-langkah sebagai berikut :Langkah-langkah penjumlahan hexadesimal : tambahkan masing-masing kolom secara desimal rubah dari hasil desimal ke hexadesimal tuliskan hasil dari digit paling kanan dari hasil hexadesimal kalau hasil penjumlahan tiap-tiap kolom terdiri dari dua digit, maka digit paling kiri merupakan carry of untuk penjumlahan kolom selanjutnya.Contoh :Desimalhexadesimal

2989 1073 + 4062BAD431 + FDE D 16 + 1 16 = 13 10 + 110 = 14 10 = E 16 A 16 + 3 16 = 10 10 + 3 10 = 13 10 =D 16 B16 + 4 16 = 1110 + 4 10 = 15 10 = F 16

b. PenguranganPengurangan bilangan hexadesimal dapat dilakukan secara sama dengan pengurangan bilangan desimal.

Contoh :Desimalhexadesimal

48331575 -325812E1 627 - CBA 16 10 (pinjam) + 1 10 - 710 = 10 10 = A 16 14 10 - 7 10 - - 1 10 (dipinjam) = 11 10 =B 16 1610 (pinjam) + 2 10 - 610 = 12 10 = C 16

1 10 1 10 (dipinjam) 0 10 = 0 16

c. PerkalianLangkah langkah : kalikan masing-masing kolom secara desimal rubah dari hasil desimal ke octal tuliskan hasil dari digit paling kanan dari hasil octal kalau hasil perkalian tiap kolol terdiri dari 2 digit, maka digit paling kiri merupakan carry of untuk ditambahkan pada hasil perkalian kolom selanjutnya.

Contoh :DesimalHexadesimal

172 27 x 1204 344 + 4644

AC 1B x 764 C 16 x B 16 =12 10 x 1110= 84 16 A16 x B16 +816 = 1010 x 1110+810=7616

AC 1B x 764 AC C16 x 116 = 1210 x 110 =1210=C16 A16 x 116 = 1010 x110 =1010=A 16

AC 1B x 764 AC + 1224 616 + C16 = 610 + 1210 = 1810 =12 16 716+A16 +116 = 710 x 1010 + 110=1810 = 1216

D. PembagianContoh :Desimalhexadesimal

27 / 4646 \ 17227- 194 189 54 54 0

1B / 1214 \ AC 10E - 1B16xA16 = 2710x1010=27010= 10E16 144 144- 1B 16 x C16 = 2710 x 10 10 = 3240 10 0 =14416

III. Konversi BilanganKonversi bilangan adalah suatu proses dimana satu system bilangan dengan basis tertentu akan dijadikan bilangan dengan basis yang alian.

Konversi dari bilangan Desimal1. Konversi dari bilangan Desimal ke binerYaitu dengan cara membagi bilangan desimal dengan dua kemudian diambil sisa pembagiannya.Contoh :

45 (10) = ..(2)45 : 2 = 22 + sisa 122 : 2 = 11 + sisa 011 : 2 = 5 + sisa 1 5 : 2 = 2 + sisa 1 2 : 2 = 1 + sisa 0101101(2) ditulis dari bawah ke atas

2. Konversi bilangan Desimal ke OktalYaitu dengan cara membagi bilangan desimal dengan 8 kemudian diambil sisa pembagiannyaContoh :385 ( 10 ) = .(8)385 : 8 = 48 + sisa 1 48 : 8 = 6 + sisa 0601 (8)

3. Konversi bilangan Desimal ke HexadesimalYaitu dengan cara membagi bilangan desimal dengan 16 kemudian diambil sisa pembagiannyaContoh :1583 ( 10 ) = .(16)1583 : 16 = 98 + sisa 15 96 : 16 = 6 + sisa 262F (16)

Konversi dari system bilangan Biner1. Konversi ke desimalYaitu dengan cara mengalikan masing-masing bit dalam bilangan dengan position valuenya.Contoh :1 0 0 11 x 2 0= 10 x 2 1= 00 x 2 2= 01 x 2 3= 810 (10)

2. Konversi ke OktalDapat dilakukan dengan mengkonversikan tiap-tiap tiga buah digit biner yang dimulai dari bagian belakang. Contoh :

11010100 (2) = (8)11 010 100

324diperjelas :100 = 0 x 2 0= 0 0 x 2 1 = 0 1 x 2 2 = 4 4Begitu seterusnya untuk yang lain.

3. Konversi ke HexademialDapat dilakukan dengan mengkonversikan tiap-tiap empat buah digit biner yang dimulai dari bagian belakang. Contoh :110101001101 0100D 4

Konversi dari system bilangan Oktal1. Konversi ke DesimalYaitu dengan cara mengalikan masing-masing bit dalam bilangan dengan position valuenya.

Contoh :12(8) = (10)2 x 8 0 = 21 x 8 1=810Jadi 10 (10)

2. Konversi ke BinerDilakukan dengan mengkonversikan masing-masing digit octal ke tiga digit biner.Contoh :6502 (8) .. = (2)

2 = 0100 = 0005 = 1016 = 110jadi 110101000010

3. Konversi ke HexadesimalDilakukan dengan cara merubah dari bilangan octal menjadi bilangan biner kemudian dikonversikan ke hexadesimal.Contoh :2537 (8) = ..(16)2537 (8) = 010101011111010101010000(2) = 55F (16)Konversi dari bilangan Hexadesimal

1. Konversi ke DesimalYaitu dengan cara mengalikan masing-masing bit dalam bilangan dengan position valuenya.

Contoh :C7(16) = (10)7 x 16 0 = 7C x 16 1= 192 199Jadi 199 (10)

2. Konversi ke OktalDilakukan dengan cara merubah dari bilangan hexadesimal menjadi biner terlebih dahulu kemudian dikonversikan ke octal.Contoh :55F (16) = ..(8)55F(16) = 010101011111(2)010101011111 (2) = 2537 (8)

Latihan :Kerjakan soal berikut dengan benar !1. Sebutkan dan jelaskan empat macam system bilangan !2. Konversikan bilangan berikut :a. 10101111(2) = .(10)b. 11111110(2) = .(8)c. 10101110101 = (16)

3. Konversi dari :a. ACD (16) = (8)b. 174 (8) = ..(2)

4. BC1 2A X

5. 245 (8) : 24 (8) =..(8) Excess-3 binary-coded decimal (XS-3), also called biased representation or Excess-N, is a numeral system used on some older computers that uses a pre-specified number N as a biasing value. It is a way to represent values with a balanced number of positive and negative numbers. In XS-3, numbers are represented as decimal digits, and each digit is represented by four bits as the BCD value plus 3 (the "excess" amount): The smallest binary number represents the smallest value. (i.e. 0Excess Value) The greatest binary number represents the largest value. (i.e. 2N+1Excess Value1)

To encode a number such as 127, then, one simply encodes each of the decimal digits as above, giving (0100, 0101, 1010).The primary advantage of XS-3 coding over BCD coding is that a decimal number can be nines' complemented (for subtraction) as easily as a binary number can be ones' complemented; just invert all bits. In addition, when the sum of two XS-3 digits is greater than 9, the carry bit of a four bit adder will be set high. This works because, when adding two numbers that are greater or equal to zero, an "excess" value of six results in the sum. Since a four bit integer can only hold values 0 to 15, an excess of six means that any sum over nine will overflow.Adding Excess-3 works on a different algorithm than BCD coding or regular binary numbers. When you add two XS-3 numbers together, the result is not an XS-3 number. For instance, when you add 1 and 0 in XS-3 the answer seems to be 4 instead of 1. In order to correct this problem, when you are finished adding each digit, you have to subtract 3 (binary 11) if the digit is less than decimal 10 and add three if the number is greater than or equal to decimal 10 (thus causing the number to wrap).

Boolean AlgebraThe logic we apply in studying digital systems is based on a definition of truth made popular by Greek philosopher Aristotle (384 BC322 BC). To Aristotle, truth had two possible values: true or false: nothing is ever both true and false at the same time; nothing ever contradicts itself; and everything is what it is. In fact, Aristotles system of truth devised four laws:1. The law of identity: which states that A is A; in other words, A = A. If its raining outside, then its raining outside. If you are in college, then you are in college. If you are happy, then you are happy.2. The law of non-contradiction: which says A is not non-A; in other words, A = not (not A). A happy person is not unhappy. An indoor pool is not outdoors. If you are a student you are not a non-student.3. The law of excluded middle: which says that either A is true or not A is true. Either you are beautiful or you are not beautiful. Either the sky is blue or it is not blue. Either your friend is pregnant or she is not pregnant.4. The law of inference: which says that you can arrive at one fact from other facts. If you are in college, and everyone in college is a student, then you are a student. If proteins are chains of amino acids, and amino acids are coded from genes, then proteins are coded from genes. If you drive drunk, and everyone who drives drunk breaks the law, then you break the law.About 2000 years after Aristotles work, an English mathematician named George Boole (18151864) turned Aristotle's logic into an algebra. 90 years later, a Masters student at MIT, Claude Shannon (19162001), proved that Booles algebra can be applied to electrical systems. This brief history is part of the reason you are studying digital systems with logic algebra today. Let's start our exploration of Boolean algebra with three scenarios. And through these three scenarios, we will demonstrate ten basic facts concerning Boolean algebra. For our first scenario, imagine that my friend Francois wants to eat a fruit and asks you to bring him an apple OR a banana. In that case you have four choices:AppleBananaFrancois's whish is grantedCOMMENT

FalseFalseFalseYou brought him neither

FalseTrueTrueYou brought him a banana

TrueFalseTrueYou brought him an apple

TrueTrueTrueYou brought him both

Table 1

In Boolean algebra, we use the plus sign (+) to mean OR. Hence, since Francois's request was Apple OR Banana, we can rewrite it as Apple + Banana. We can also rewrite your four choises as False+ False=False

False+ True=True

True+ False=True

True+ True=True

One further step that we take is to let TRUE = 1 and FALSE = 0, which allows us to further rewrite your four choices as Axiom 1.0+ 0=0

Axiom 2.0+ 1=1

Axiom 3.1+ 0=1

Axiom 4.1+ 1=1

Notice that in Boolean algebra 1 + 1 = 1. This is because 1 stands for TRUE and TRUE + TRUE = TRUE. What this also means is that Boolean numbers and binary numbers arent the same. In binary arithmetic 1 + 1 = 10. Binary numerals is just a translation of the decimal system; Boolean algebra is about logic. Anyway, lets continue to our second scenario.

Now lets apply the same Francois situation to another operator: AND. This time Francois wants to make smoothie and asks you to bring him an apple AND a banana. You still have four choices and each one of them will affect Francoiss wish.AppleBananaFrancois's whish is grantedCOMMENT

FalseFalseFalseYou brought him neither

FalseTrueFalseYou brought him a banana

TrueFalseFalseYou brought him an apple

TrueTrueTrueYou brought him both

Table 2

Because Francois specified that he needed an apple AND a banana, you will only satisfy his command by bringing both. For the AND operator, we use the symbol in Boolean algebra. Hence, we can rewrite your four choices asAxiom 5.0 0=0

Axiom 6.01=0

Axiom 7.10=0

Axiom 8.11=1

If you think of the AND operator as the multiplication operator in normal algebra, then you are right on the money. They are not the same, but they work the same way. Now lets proceed to our third and final scenario. Imagine this time that Francois is allergic to apple and tells you that no matter what you do, do NOT bring apple into the house. In this final case you only have two choices which of course will affect Francois.AppleFrancoiss wish is granted

FalseTrue

TrueFalse

Table 3

For the NOT operator we use the apostrophe (). Hence, we rewrite your two choices asAxiom 9.0=1in Boolean algebra 0 is the opposite of 1.

Axiom 10.1 =0in Boolean algebra 1 is the opposite of 0.

The ten choices you just made regarding the different situations of Francois are known as the basic facts (or axiom if you want to be fancy) of Boolean algebra. Most textbooks write these facts as complementary pairs to show socalled duality. For mere mortals like you and me, this complementary to show duality business simply means that in Boolean algebra if a statement is true then its opposite is also true. Your professor may not like that I am telling you that complementary means opposite, but trust me thats all you need to know. Below are the five opposite pairs. (I call them complements just to stay on the good side of your professor. But feel free to call them opposites.)AND ()complementOR(+)

AcomplementA

1complement0

A = 1complementA = 0

A = 0complementA = 1

To show you how this complement stuff works, I have rearranged the ten axioms in pair of opposites. Notice that I get the opposite of an expression by swapping 0 and 1, and by swapping and +.1 + 1 = 1complement0 0 = 0

1 + 0 = 1complement0 1 = 0

0 + 1 = 1complement1 0 = 0

0 + 0 = 0complement1 1 = 1

0complement1

1complement0

As you will find out soon, the duality of the complements (i.e., the fact that the opposite of a true statement is also true) makes it easy to simplify expressions in Boolean algebra. But before I show you, here is a list of convenient truths (textbooks call them theorems) that we get from manipulating the basic facts (axioms) above. Dont try to memorize the list. If you cant see why something in the list is true, just replace the letters with 0s and 1s and they will make sense; for example, A + 0 = A is true because if you replace A with 1 you get 1 + 0 = 1 and if you replace A with 0 you get 0 + 0 = 0. You see how easy it is? Theorem 1.A + 0 = A A 1 = AIdentities

Theorem 2.A + 1 = 1A 0 = 0Null elements

Theorem 3.A + A = AA A = ASameness/Idempotency

Theorem 4.A + A = 1 A A = 0Complements

Theorem 5.( A ) = A Involution

Theorem 6.A + B = B + A A B = B ACommutative

Theorem 7.A + (B + C) = (A + B) + C A (B C) = (A B) CAssociative

Theorem 8.A (B + C) = A B + A C A + B C = (A + B) (A + C) Distributive

Theorem 9.(A + B) = A B (A B) = A + BDeMorgans theorem

Theorem 10.A + A B = A + BA (A + B) = A BDeMorgans theorem

Theorem 11.A + A B = A A (A + B) = AAbsorption/Covering

Theorem 12.A B + A B = A (A + B) (A + B) = ACombining

Theorem 13.A B + A C + B C = A B + A C (A+B) (A+C) (B+C) = (A+B) (A+C)Consensus

I know that was a long list, but its a good idea to at least read through it once. Nobody memorizes this thing. As you simplify more and more Boolean expressions, they will come to you. Its as I said: we derive all of them from the axioms. Now, let me show you an example where we use the theorems to simplify an expression.

Given F = A B + A B + A B, simplify F.Solution:F=A B + A (B + B)we factor out the A using the distributive law (theorem 8)

=A B + A (1)from theorem 4 we know that B + B = 1 (complements)

=A B + Afrom theorem 1 we know that A 1 = A (identity)

=A + B finally, from theorem 10 we know that A + A B = A + B

Therefore F = A + B! Here are a few more examples. You can do them first and then check your work against mine for reinforcement.

Show that A + B C = (A + B) (A + C) is true.Solution:We will simplify the right side of the equation to make it look like the left side.A + B C (A + B) (A + C)

A A + A C + B A + B CAfter expanding (think FOIL from algebra class)

A + A C + B A + B CFrom theorem 3 we know that A A = A (Idempotency)

A (1 + C + B) + B CAfter factoring out the A (just like in normal algebra)

A (1) + B CFrom theorem 2 we know that 1 plus anything is 1

A + B CFrom theorem 1 we know that A 1 = A (Identity)

Voila!

Logic NOTThe NOT gate is an electronic circuit that performs negation. The output of the NOT Boolean operator is true when the input is false, and the output is false when the input is true. Hence, all NOT does is take an operand and negate it. For the sake of argument let us analyze a sentence that uses the word NOT: Good is NOT evil. A philosopher will tell us the absence of good does not necessitate the presence of evil. However that may be, our concern here is not meaning. Our concern is the statement. As we show in Table 1, the operand Good can be in two possible states: either it is present or it is not present. As such, we must evaluate the statement for two possibilities. We show the complete evaluation in Table 2. MorePossibilitiesGoodCOMMENT

Case 1FalseGood is absent

Case 2TrueGood is present

Table 1: possible input conditions

GoodEvilCOMMENT

FALSETRUEGood is absent

TRUEFALSEGood is present

Table 2: Truth Table of complete evaluation

It is typical in engineering to use 1 instead of TRUE and 0 instead of FALSE. Therefore we rewrite the data from Table 2 in Table 3 accordingly. GoodEvilCOMMENT

01Good is absent

10Good is present

Table 3: Truth Table of complete evaluation

For greater detail on the logic significance of the word NOT, read the Boolean Algebra article. Physical ImplementationIn order to apply the principles of Boolean algebra to create real machines that can think and make decisions, we have had to find ways to physically implement the logic operators AND, OR, NOT, etc. To that end, modern day engineering uses transistor networks called logic gates. Hence, a logic gate is actually a group of transistors so arranged as to behave as a Boolean operator. From a circuit complexity perspective, the most basic logic gate is the NOT gate (aka the Inverter). The NOT gate is made of two transistors, as shown in Figure 1. In theory a NOT gate is really just one transistor. But in practice microchip manufacturers use a pair of transistors to construct the NOT gate.

Figure 1: Interactive transistor circuit of the NOT logic operator

TransistorsThe use of transistors to build logic gates is quite modern. Before transistors we used other devices, such as vacuum tubes (aka thermionic valves). And very soon we may use DNA, or some other abundant material. There are many types of transistors. Our circuit in Figure 1, for example, uses complementary metaloxide semiconductor (CMOS) technology. Our choice of CMOS is arbitrarily based on the fact that CMOS is by far the dominant technology in use today. The dominance is due to how well CMOS performs in all the important categories: fabrication cost, packing density, loading capacity (i.e. fanout), operational speed (i.e. propagation delay), noise margin, and power dissipation (i.e. green technology). There is of course more to transistors than can be presented here; especially since transistors are used for more than just digital systems. And so we refer you to any good micro-electronics textbook. Alternate DesignBelow we show three additional typical constructions of the NOT gate. Each of the constructions presents specific conveniences to designers. If you are very new to digital systems design, you may not understand the importance of the figures below. Still, we include them in this article for the people who may need them.

LogicThe AND gate is an electronic circuit that performs logical conjunction. The output of the AND Boolean operator is true only when all the inputs are true. Otherwise, the output is false. Through a short and simple analysis you can determine what a statement that uses the word AND is truly saying. We illustrate with the following statement: Water is the product of Hydrogen AND Oxygen. The statement tells us that the existence of Water depends on the existence of two objects: Hydrogen and Oxygen. Now each of these two objects can be in two possible definitive states: either an object exists or it does not exist. Consequently we have four possible conditions, which we list in Table 1 below by row. MoreConditionHydrogen existsOxygen existsCOMMENT

Case 1FalseFalseneither exists

Case 2FalseTrueonly oxygen exists

Case 3TrueFalseonly hydrogen exists

Case 4TrueTrueboth exist

Table 1: possible input conditions

You can list the possibilities however you want; you will still end up with four different cases. Try it if you want. Once we list the possible input conditions, we can proceed by adding a column for water as in Table 2.Hydrogen existsOxygen existsWater exists

FalseFalse

FalseTrue

TrueFalse

TrueTrue

Table 2: Empty water columnTo fill the column for Water, we simply do exactly as the original statement says: Water is the product of Hydrogen AND Oxygen. So for example, the first row for water must be false since the statement did not say we can produce water out of nothing. We fill all the rows in Table 3 below. Notice that water is shown to exist only when both Hydrogen and Oxygen exist. Hydrogen existsOxygen existsWater existsComments

FalseFalseFalsecan't make water out of nothing

FalseTrueFalseoxygen alone is not water

TrueFalseFalsehydrogen alone is not water

TrueTrueTruenow we have water

Table 3: Truth Table of Water = Hydrogen AND OxygenLogic ANDThe AND gate is an electronic circuit that performs logical conjunction. The output of the AND Boolean operator is true only when all the inputs are true. Otherwise, the output is false. Through a short and simple analysis you can determine what a statement that uses the word AND is truly saying. We illustrate with the following statement: Water is the product of Hydrogen AND Oxygen. The statement tells us that the existence of Water depends on the existence of two objects: Hydrogen and Oxygen. Now each of these two objects can be in two possible definitive states: either an object exists or it does not exist. Consequently we have four possible conditions, which we list in Table 1 below by row. MoreConditionHydrogen existsOxygen existsCOMMENT

Case 1FalseFalseneither exists

Case 2FalseTrueonly oxygen exists

Case 3TrueFalseonly hydrogen exists

Case 4TrueTrueboth exist

Table 1: possible input conditions

You can list the possibilities however you want; you will still end up with four different cases. Try it if you want. Once we list the possible input conditions, we can proceed by adding a column for water as in Table 2.Hydrogen existsOxygen existsWater exists

FalseFalse

FalseTrue

TrueFalse

TrueTrue

Table 2: Empty water columnTo fill the column for Water, we simply do exactly as the original statement says: Water is the product of Hydrogen AND Oxygen. So for example, the first row for water must be false since the statement did not say we can produce water out of nothing. We fill all the rows in Table 3 below. Notice that water is shown to exist only when both Hydrogen and Oxygen exist. Hydrogen existsOxygen existsWater existsComments

FalseFalseFalsecan't make water out of nothing

FalseTrueFalseoxygen alone is not water

TrueFalseFalsehydrogen alone is not water

TrueTrueTruenow we have water

Table 3: Truth Table of Water = Hydrogen AND Oxygen

TransistorsThe use of transistors to build logic gates is quite modern. Before transistors we used other devices, such as vacuum tubes (aka thermionic valves). And very soon we may use DNA, or some other abundant material. There are many types of transistors. Our circuits in figures 1, 2 and 3, for example, use complementary metal-oxide semiconductor (CMOS) technology. Our choice of CMOS is arbitrarily based on the fact that CMOS is by far the dominant technology in use today. The dominance is due to how well CMOS performs in all the important categories: fabrication cost, packing density, loading capacity (i.e. fan-out), operational speed (i.e. propagation delay), noise margin, and power dissipation (i.e. green technology).

There is of course more to transistors than can be presented here; especially since transistors are used for more than just digital systems. And so we refer you to any good micro-electronics textbook. Alternate DesignBelow we show three additional typical constructions of the AND gate. Each of the constructions presents specific conveniences to designers. If you are very new to digital systems design, you may not understand the importance of the figures below. Still, we include them in this article for the people who may need them.

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