Digital Logic

Basic Logic Gates

Name	Graphical Symbol	Algebraic Expression
AND		F=A.B OR F=AB
OR		F=A+B
NOT		F=A’ OR F=!A
NAND		F=(A.B)’
NOR		F=(A+B)’
XOR		F=A.B’+A’.B

Combinational of circuits that can be define in three ways:

1) Truth table

2) Graphical symbols

3) Boolean equations

Boolean Equation Forms

SOP  -  combination of input values that produce 1s is convert into equivalent variables , AND together then OR with other combination variables with the same output.

POS - input combinations that produce 0 in sum terms (OR variables) are AND together.

Simplification of Boolean equation

1) Laws of Boolean Algebra

2)Karnaugh Map

Created by:

NUR FALAHI NABIHAH BINTI AHMAD SHAFRI
B031310253

BASIC LOGIC GATES

·     
      A logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation on one or more logical inputs, and produces a single logical output. 

TRUTH TABLE

A truth table is a mathematical table used in logic specifically in connection with Boolean algebra, boolean functions, and propositional calculus to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables.

AND gates

·    
      The AND gate is a basic digital logic gate that implements logical conjunction it behaves according to the truth table to the right. A HIGH output (1) results only if both the inputs to the AND gate are HIGH (1). If neither or only one input to the AND gate is HIGH, a LOW output results. In another sense, the function of AND effectively finds the minimum between two binary digits, just as the OR function finds the maximum. Therefore, the output is always 0 except when all the inputs are 1s. Example :

OR gates

·       
      The OR gate is a digital logic gate that implements logical disjunction - it behaves according to the truth table to the right. A HIGH output (1) results if one or both the inputs to the gate are HIGH (1). If neither input is HIGH, a LOW output (0) results. In another sense, the function of OR effectively finds the maximum between two binary digits, just as the complementary AND function finds the minimum. Example :

NOT gates

·        
      NOT gates or inverters have a single bit input and a single bit of output. Example :

NAND gates

·     Digital logic gate with two or more inputs and one output with behavior that is the opposite of an AND gate. The output of a NAND gate is true when one or more, but not all, of its inputs are false. If all of a NAND gate's inputs are true, then the output of the NAND gate is false. 
      Example :

XOR gates

·       
      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. Example :

 

Created by:

Ahmad Nur Fathin Izzat b. Azli

B031310273

      
      


      Karnaugh Map ( K-Map)

In this subtopic, we learn how to simplified Boolean expressions. The only limitation is that it will be ineffective for more than 4 inputs.

Example for K-Map:

Truth Table:

A	B	C	Minterm
0	0	0	A’B’C’
0	0	1	A’B’C
0	1	0	A’BC’
0	1	1	A’BC
1	0	0	AB’C’
1	0	1	AB’C
1	1	0	ABC’
1	1	1	ABC

From the truth table above, we make a K-Map as shown below:

BC        A	B’C’ (00)	B’C (01)	BC (11)	BC’ (10)
A’(0)	0	1	1	0
A(1)	0	1	1	1

 

Created by:

AZIRA BT HAMDAN

B031310453

 

K – MAP (KARNAUGH MAP)

K-MAP GROUPING

K map is used to simplify Boolean expression. It enable us to simplify the Boolean expression quickly without taking too much time. It can be used for 2 input, 3 input and 4 input only. More or less than that make the k map malfunction.

example :

Truth Table(3 input)

A	B	C	F
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

First construct a table like this. (2^3)

Then fill up the table according to the truth table given.

But remember that the most important rule of constructing the table

Now find the largest group of the same output.

After that try to examine the result by checking which variable between A, B and C that doesn’t changes.

So we can conclude that for the group the simplified expression is

Then find the other group of 1.

After that, do the same thing. Find the variable that does not change in the group.

Therefore we can conclude for this group is

So combine the result for both group that we get.

The simplified Boolean expression for the truth table is:

F = AB + C

The last thing we have to do is to test the simplified Boolean expression with the truth table whether is satisfy the table.

Finally we can conclude that the F = AB + C is the simplified Boolean expression for the truth table given

Created by:

EDDY FAZZIRRULLAH BIN AZMI
B031310313

Universal Gates

Any combination of basic gate like AND, OR, and NOT are called universal gates. NAND and NOR are such of examples.

A NAND gate is an inverted AND gate. It has the following truth table:

Q = NOT( A AND B )

Truth Table
Input A	Input B		Output Q
0	0		1
0	1		1
1	0		1
1	1		0

A NOR gate is simply an inverted OR gate. Output is high when neither input A nor input B is high:

 Q = NOT (A OR B)    

= NOT{NOT[NOT(A AND A) AND NOT(B AND B)]}

Truth Table Input A Input B Output Q 0 0 1 0 1 0 1 0 0 1 1 0

Created by:
NOOR AZREEANI BINTI AZMAN
B031310566