Modul Outcome 1 Part 2

8/7/2019 Modul Outcome 1 Part 2

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Computer

System

(CSC 1033)

Outcome 1:

Evaluate data

representation

and manipulation

within a

computer.

Criteria 1.5


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Computer Systems (CSC 1033)

Outcome 1: Evaluate data representation and manipulation within a computer

Page 2

TABLE OF CONTENTNO. TOPICS PAGE

1. 1.5 Specify different types of logic gates 3

2. 1.6 Apply different ways to perform arithmetical operations of logic gates. 7

Prepared by:

Pn. Noor zilawati Binti Sabtu

Science & Quantitative Department, KPM Beranang

Updated - 17 JAN 2010


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1.5 Specify different types of logic gates.Definition of Gates

Keep in mind that computers work on an electrical flow where a high voltage is considered a 1

and a low voltage is considered a 0. Using these highs and lows, data are represented.

Electronic circuits must be designed to manipulate these positive and negative pulses into

meaningful logic.

Logic gates are the building blocks of digital circuits. Combinations of logic gates form circuits

designed with specific tasks in mind. For example, logic gates are combined to form circuits to

add binary numbers (adders), set and reset bits of memory (flip flops), multiplex multiple

inputs, etc.

Definition of Truth Table

A truth table is a table that describes the behavior of a logic gate. It lists the value of the output

for every possible combination of the inputs and can be used to simplify the number of logic

gates and level of nesting in an electronic circuit. In general the truth table does not lead to an

efficient implementation; a minimization procedure, using Karnaugh maps, the Quine

McCluskey algorithm or an heuristic algorithm is required for reducing the circuit complexity.


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OR Gate (X , Y = X + Y)

The OR gate has two input lines and one output line. Basically, if either or both of the inputs

are a 1, the resulting output value is a 1. Note in the truth table, the only time the output is 0 is

when both inputs are 0.

OR Gate Truth Table

Input

Output

A B

0 0 0

0 1 1

1 0 1

1 1 1

EXCLUSIVE OR, XOR Gate (X , Y = X + Y)

The X in the XOR gate stands for "exclusive." This means that the output from this gate will be

a 1 ONLY when one or the other of the inputs is a 1. Notice in the truth table that the output is

a 0 if both the inputs are 1 or 0. In other words, this is an either-or gate.


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XOR Gate Truth Table

Input

Output

A B

0 0 0

0 1 1

1 0 1

1 1 0

NOT - AND, NAND Gate (X , Y = [X . Y])

The Logic NAND Gate is a combination of the digital logic AND gate with that of an inverter or

NOT gate connected together in series. The NAND (Not - AND) gate has an output that is

normally at logic level "1" and only goes "LOW" to logic level "0" when ALL of its inputs are at

logic level "1". The Logic NAND Gate is the reverse or "Complementary" form of the AND gatewe have seen previously.

The logic or Boolean expression given for a logic NAND gate is that for Logical Addition, which is

the opposite to the AND gate, and which it performs on the complements of the inputs. The

Boolean expression for a logic NAND gate is denoted by a single dot or full stop symbol, (.) with

a line or Overline, ( ) over the expression to signify the NOT or logical negation of the NAND

gate giving us the Boolean expression of: A.B = Q. Then we can define the operation of a 2-

input logic NAND gate as being: "If either A or B are NOT true, then Q is true"

A B Q = [A.B]

0 0 1

0 1 1

1 0 1

1 1 0


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NOT - OR, NOR Gate (X , Y = [X + Y])

The Logic NOR Gate or Inclusive-NOR gate is a combination of the digital logic OR gate with that of an

inverter or NOT gate connected together in series. The NOR (Not - OR) gate has an output that is

normally at logic level "1" and only goes "LOW" to logic level "0" when ANY of its inputs are at logic level

"1". The Logic NOR Gate is the reverse or "Complementary" form of the OR gate we have seen

previously.

The logic or Boolean expression given for a logic NOR gate is that for Logical Multiplication

which it performs on the complements of the inputs. The Boolean expression for a logic NOR

gate is denoted by a plus sign, (+) with a line or Overline, ( ) over the expression to signify the

NOT or logical negation of the NOR gate giving us the Boolean expression of: A+B = Q. Then wecan define the operation of a 2-input logic NOR gate as being: "If both A and B are NOT true,

then Q is true"

A B Q = [A+B]

0 0 1

0 1 01 0 0

1 1 0


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1.6 Apply different ways to perform arithmetical operations of logic gates.Basic Rules of BOOLEAN ALGEBRA

The basic rules for simplifying and combining logic gates are called Boolean algebra in honour of

George Boole (1815 1864) who was a self-educated English mathematician who developed

many of the key ideas. The following set of exercises will allow you to rediscover the basic rules:

Example 1:

Consider the AND gate where one of the inputs is 1. By using the Truth table, investigate

the possible outputs and hence simplify the expression x 1.

Solution :

From the truth table for AND, we see that if x is 1 then 1 1 = 1, while if x is 0

then 0 1 = 0. This can be summarized in the rule that x 1 = x,

Example 2:

Consider the AND gate where one of the inputs is 0. By using the truth table, investigate thepossible outputs and hence simplify the expression x 0.

Solution:

From the truth table for AND, we see that if x is 1 then 1 0 = 0, while if x is 0 then 0 0 = 0. This

can be summarized in the rule that x 0 = 0


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Rules of BOOLEAN ALGEBRA

(1a) x y = y x

(1b) x + y = y + x

(2a) x (y z) = (x y) z

(2b) x + (y + z) = (x + y) + z

(3a) x (y + z) = (x y) + (x z)

(3b) x + (y z) = (x + y) (x + z)

(4a) x x = x

(4b) x + x = x

(5a) x (x + y) = x

(5b) x + (x y) = x

(6a) x x0 = 0(6b) x + x0 = 1

(7) (x0)0 = x

(8a) (x y)0 = x0 + y0

(8b) (x + y)0 = x0 y0

These rules are a direct translation into the notation of logic gates of the rules derived in the

package Truth Tables and Boolean Algebra. We have seen that they can all be checked by

investigating the corresponding truth tables. Alternatively, some of these rules can be derived

from simpler identities derived in this package.

Example 3:Show how rule (5a) can be derived from the basic identities derived earlier.

Solutionx (x + y) = x x + x y using (3a)

= x + x y ..using (4a)

= x (1 + y) ..using (3a)= x 1 .using Example 1

= x .as required

The examples above have all involved at most two inputs. However, logic gates can be puttogether to join an arbitrary number of inputs. The Boolean algebra rules of the table are

essential to understand when these circuits are equivalent and how they may be simplified.


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Example 4:

Let us consider the circuits which combine three inputs via AND gates. Two different ways of

combining them are

However, rule (2a) states that these gates are equivalent. The order of taking AND gates is not

important. This is sometimes drawn as a three (or more!) input AND gate but really this just

means repeated use of AND gates as shown below.

Implementing Circuits from Boolean Expression

If the operation of a circuit is defined by a Boolean expression, a logic-circuit diagram can he

implemented directly from that expression. Suppose that we wanted to construct a circuit

whose output is y = AC+BC' + A'BC.

This Boolean expression contains three terms (AC, BC', A'BC), which are ORed together. This

tells us that a three-input OR gate is required with inputs that are equal to AC, BC', and A'BC,

respectively.

Each OR-gate input is an AND product term, which means that an AND gate with appropriate

inputs can be used to generate each of these terms. Note the use of INVERTERs to produce the

A' and C' terms required in the expression.


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Evaluating Logic Circuits Output

Once the Boolean expression for a circuit output has been obtained, the output logic level can

be determined for any set of input levels. These are two examples of the evaluating logic circuit

output:

Example 5:

Let A=0, B=1, C=1, D=1; X?

Solution:

X = A'BC (A+D)'

= 0'*1*1* (0+1)'

= 1 *1*1* (1)'

= 1 *1*1* 0

X = 0

Example 6:

Let A=0, B=0, C=1, D=1; X?

Solution:

X = [D+ ((A+B)C)'] * E

= [1 + ((0+0)1 )'] * 1

= 1 + (0*1)'] * 1

= [1+ 0'] *1

= [1+ 1] * 1

X = 1

In general, the following rules must always be followed when evaluating a Boolean expression:

1. First, perform all inversions of single terms; that is, 0 = 1 or 1 = 0.

2. Then perform all operations within parentheses.

3. Perform an AND operation before an OR operation unless parentheses indicate

otherwise.

4. If an expression has a bar over it, perform the operations of the expression first and

then invert the result.


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Describing Logic Circuits Algebraically

Any logic circuit, no matter how complex, may be completely described using the Boolean

operations, because the OR gate, AND gate, and NOT circuit are the basic building blocks of

digital systems. This is an example of the circuit using Boolean expression:

Whenever an INVERTER is present in a logic-circuit diagram, its output expression is simply

equal to the input expression with a prime (') over it.


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DeMorgans Theorem

DeMorgan's theorems are extremely useful in simplifying expressions in which a product or

sum of variables is inverted. The two theorems are:

(16). (x+y)' = x' * yTheorem (16) says that when the OR sum of two variables is inverted, this is the same as

inverting each variable individually and then ANDing these inverted variables.

(17). (x*y)' = x' + yTheorem (17) says that when the AND product of two variables is inverted, this is the same as

inverting each variable individually and then ORing them.

Three Variables DeMorgan's Theorem

(18).. (x+y+z)' = x' * y' * z'

(19)(xyz)' = x' + y' + z'

For: (16)..(x+y)' = x' * y'


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For: (17)..(x*y)' = x' + y'

Modul Outcome 1 Part 2

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