Logic Gates
Introduction
Section titled “Introduction”Let’s introduce a useful convention LOW, 0
and false
have the same logic meaning, as well as HIGH, 1
and true
.
Bool Algebra
Section titled “Bool Algebra”Axioms | Single variable theorems |
---|---|
0 ⋅ 0 = 0 | x ⋅ 0 = 0 |
1 + 1 = 1 | x + 1 = 1 |
1 ⋅ 1 = 1 | x ⋅ 1 = x |
0 + 0 = 0 | x + 0 = x |
0 ⋅ 1 = 1 ⋅ 0 = 0 | x ⋅ x = x |
0 + 1 = 1 + 0 = 1 | x + x = x |
if x = 0 , then !x = 1 | x ⋅ !x = 0 |
if x = 1 , then !x = 0 | x + !x = 1 |
1 = !!1 and 0 = !!0 | x = !!x |
Please Note: Double negation is often used with a couple of overline signs.
Duality Principle
Section titled “Duality Principle”Given a logic expression, the dual is obtained by swapping:
- summations with multiplications (and vice-versa)
0
with1
(and vice-versa)
Main Gates
Section titled “Main Gates”AND gate is a logic gate that implements logical conjunction (∧).
If one or both the inputs are false
, the output is false
.
If both inputs are true
, the output is true
.
Truth table:
A | B | A ⋅ B |
---|---|---|
0 | 0 | 0 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
OR gate
Section titled “OR gate”OR gate is a logic gate that implements logical disjunction (∨).
A true
output results if one or both the inputs are true
.
If neither input is true
, the output is false
.
Truth table:
A | B | A + B |
---|---|---|
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 1 |
NOT gate
Section titled “NOT gate”NOT gate is a logic gate that implements logical negation (¬).
It is more commonly referred with the exclamation mark !
before che operator: !A
means logical negation of A
.
Truth table:
A | ¬ A |
---|---|
0 | 1 |
1 | 0 |
Compound Gates
Section titled “Compound Gates”NOR gate
Section titled “NOR gate”NOR gate is the result of the negation of the OR operator. It can also be seen as the inverse of an AND gate.
A true
output results if both the inputs are false
.
A false
output results if one or both the inputs are true
.
Truth table:
A | B | ¬ (A ∨ B) |
---|---|---|
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 0 |
NAND gate
Section titled “NAND gate”NAND gate is the result of the negation of the NAND operator. It can also be seen as the inverse of an OR gate.
A true
output results if one or both the inputs are false
.
A false
output results if both the inputs are true
.
Truth table:
A | B | ¬ (A ∧ B) |
---|---|---|
0 | 0 | 1 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
XOR gate
Section titled “XOR gate”XOR gate is an Exclusive OR gate that performs exclusive disjunction (⊕).
The output of an XOR gate is true
only when exactly one of its inputs is true
.
If inputs are both false
or both true
the output is false
.
Truth table:
A | B | A ⊕ B |
---|---|---|
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
XNOR gate
Section titled “XNOR gate”XNOR gate is the logical complement of XOR gate (⨀).
The output of an XNOR gate is false
only when exactly one of its inputs is true
.
If inputs are both false
or both true
the output is true
.
Truth table:
A | B | A ⨀ B |
---|---|---|
0 | 0 | 1 |
0 | 1 | 0 |
1 | 0 | 0 |
1 | 1 | 1 |
Conclusion
Section titled “Conclusion”Useful links:
- Logic Gates Symbols
- wumbo.net
- AND sign, DOT operator
- OR sign, also on fileformat.info
- negation sign, also on fileformat.info
- XOR sign
- XNOR