Boolean algebra and logical gates

Logical gates overview

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Logical gates are the basic of the computerized world and is based on the digital values of the binary numbers 0 and 1. The implementation of the logical gates are performed by the rules of the Boolean algebra, and based on the combinations of the operations OR, AND and NOT. The specific gate operation is attained by using diodes or transistors that acts like a switch 0 is off (0 Volt) and 1 is on (5 Volt).

Operation	Logical symbol	Switch operation	Electronics
Logical OR and AND gates description
OR		0
AND

Logical gates examples 1, 2 and 3

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Num	X Y Z	XY'	Output
F(X, Y,Z) = XY' + Z
1	0 0 0	0	0
2	0 0 1	0	1
3	0 1 0	0	0
4	0 1 1	0	1
5	1 0 0	1	1
6	1 0 1	1	1
7	1 1 0	0	0
8	1 1 1	0	1

Num	A B	Output
1	0 0	1
2	0 1	1
3	1 0	0
4	1 1	0




Output

From the output result we can see that the system can be simplified to the equivalent form.

Num	A B	Output
F(A, B) = A XOR B
1	0 0	0
2	0 1	1
3	1 0	1
4	1 1	0



Output

This system is the equivalent of the XOR gate.

Logical gates example 4

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The truth table at points D, E, F, G, H and K of the system described above are.

A	B	C	D	E	F	G	H	K
0	0	0	0	0	0	0	0	0
0	0	0	1	0	0	1	0	0
0	0	1	0	0	0	0	0	0
0	0	1	1	0	0	1	0	0
0	1	0	0	0	1	1	0	0
0	1	0	1	0	0	1	0	0
0	1	1	0	0	1	1	0	0
0	1	1	1	0	0	1	0	0
1	0	0	0	0	0	0	0	0
1	0	0	1	0	0	1	1	1
1	0	1	0	0	0	0	0	0
1	0	1	1	0	0	1	1	1
1	1	0	0	1	1	1	1	1
1	1	0	1	1	0	1	1	1
1	1	1	0	0	1	1	1	1
1	1	1	1	0	0	1	1	1

Port	Value	Notes
E
F
G		D + F
H		A‧G
K		E + H

Logical gates example 5

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The truth table at points D, E, F, G, H, I, J, and K of the system described above are.

A	B	C	D	E	F	G	H	I	J	K
0	0	0	1	1	1	0	0	1	1	0
0	0	1	1	1	0	1	0	1	0	0
0	1	0	1	0	1	1	0	1	0	0
0	1	1	1	0	0	1	0	1	0	0
1	0	0	0	1	1	1	0	1	0	0
1	0	1	0	1	0	1	0	1	0	0
1	1	0	0	0	1	1	0	1	0	0
1	1	1	0	0	0	1	1	0	1	1

Port	Value	Notes
D		NOT Buffer
E		NOT Buffer
F		NOT Buffer
G		De Morgan's theorem
H
I
J
K

Logical gates example 6

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The truth table at points C, D, E, F and G of the system described above are.

A	B	C	D	E	G
0	0	0	1	0	0
0	1	0	1	1	1
1	0	0	1	1	1
1	1	1	1	0	0

Port

Value

(D equals 1 see above)

Logical gates example 7

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The truth table at points D, E, F, G, H and K of the system described above are.

A	B	C	D	E	F	G	H	K
0	0	0	0	0	0	0	0	0
0	0	1	0	0	0	1	1	1
0	1	0	1	1	0	0	1	1
0	1	1	1	1	1	1	0	1
1	0	0	1	0	0	0	1	0
1	0	1	1	0	0	0	0	0
1	1	0	0	0	0	0	0	0
1	1	1	0	0	1	0	1	1

Port	Value	Notes
D
E
F
G
H
K

Draw logical gate circuit from Boolean expression example 11

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If we have a logical expression such as		we can draw the equivalent
logical gate circuit by the following steps:

Divide the expression into 2 expressions separated by the OR (+) operator marked by X and Y.

this is equivalent to the gate:

The X expression can be divided into AND gate after simplifying:

this expression is the gate:

Perform the same process on the Y expression to get an AND of 2 expressions:

this expression is the gate:

Combine the gates of section 1, 2 and 3 to get the final scheme of the complete gate: