Logic Gate Truth Tables
As well as a standard Boolean Expression, the input and output information of any Logic Gate
or circuit can be plotted into a table to give a visual representation of the switching function of the system and this is
commonly called a Truth Table. A logic gate truth table shows each possible input combination to the gate
or circuit with the resultant output depending upon the combination of these input(s).
For example, consider a single 2-input logic circuit with input variables labelled as
A and B. There are "four" possible input combinations or
22 of "OFF" and "ON" for the two inputs. However, when dealing with Boolean expressions
and especially logic gate truth tables, we do not general use "ON" or "OFF" but instead give them bit values which represent
a logic level "1" or a logic level "0" respectively.
Then the four possible combinations of A and B for a
2-input logic gate is given as:
- Input Combination 1. - "OFF" - "OFF" or ( 0, 0 )
- Input Combination 2. - "OFF" - "ON" or ( 0, 1 )
- Input Combination 3. - "ON" - "OFF" or ( 1, 0 )
- Input Combination 4. - "ON" - "ON" or ( 1, 1 )
Therefore, a 3-input logic circuit would have 8 possible input combinations or 23 and a 4-input logic
circuit would have 16 or 24, and so on as the number of inputs increases. Then a logic circuit with
"n" number of inputs would have 2n possible input combinations
of both "OFF" and "ON". In order to keep things simple to understand, we will only deal with simple 2-input logic gates,
but the principals are still the same for gates with more inputs.
The Truth tables for a 2-input AND Gate,
a 2-input OR Gate and a NOT Gate are given as:
2-input AND Gate
For a 2-input AND gate, the output Q is true if BOTH
input A "AND" input B are both true, giving the Boolean
Expression of: ( Q = A and B ).
| Symbol | Truth Table | ||
 |
A | B | Q |
| 0 | 0 | 0 | |
| 0 | 1 | 0 | |
| 1 | 0 | 0 | |
| 1 | 1 | 1 | |
| Boolean Expression Q = A.B | Read as A AND B gives Q | ||
Note that the Boolean Expression for a two input AND gate can be written as:
A.B or just simply AB without the decimal point.
2-input OR (Inclusive OR) Gate
For a 2-input OR gate, the output Q is true if EITHER
input A "OR" input B is true, giving the
Boolean Expression of: ( Q = A or B ).
| Symbol | Truth Table | ||
 |
A | B | Q |
| 0 | 0 | 0 | |
| 0 | 1 | 1 | |
| 1 | 0 | 1 | |
| 1 | 1 | 1 | |
| Boolean Expression Q = A+B | Read as A OR B gives Q | ||
NOT Gate
For a single input NOT gate, the output Q is ONLY true
when the input is "NOT" true, the output is the inverse or complement of the input giving the
Boolean Expression of: ( Q = NOT A ).
| Symbol | Truth Table | ||
 | A | Q | |
| 0 | 1 | ||
| 1 | 0 | ||
| Boolean Expression Q = NOT A or A | Read as inverse of A gives Q | ||
The NAND and the NOR Gates are a combination of the
AND and OR Gates with that of a NOT Gate
or inverter.
2-input NAND (Not AND) Gate
For a 2-input NAND gate, the output Q is true if BOTH
input A and input B are NOT true, giving the Boolean Expression of:
( Q = not(A and B) ).
| Symbol | Truth Table | ||
 |
A | B | Q |
| 0 | 0 | 1 | |
| 0 | 1 | 1 | |
| 1 | 0 | 1 | |
| 1 | 1 | 0 | |
| Boolean Expression Q = A .B | Read as A AND B gives NOT-Q | ||
2-input NOR (Not OR) Gate
For a 2-input NOR gate, the output Q is true if BOTH
input A and input B are NOT true, giving the Boolean Expression of:
( Q = not(A or B) ).
| Symbol | Truth Table | ||
| A | B | Q |
| 0 | 0 | 1 | |
| 0 | 1 | 0 | |
| 1 | 0 | 0 | |
| 1 | 1 | 0 | |
| Boolean Expression Q = A+B | Read as A OR B gives NOT-Q | ||
As well as the standard logic gates there are also two special types of logic gate function called
an Exclusive-OR Gate and an Exclusive-NOR Gate. The actions of
both of these types of gates can be made using the above standard gates however, as they are widely used functions,
they are now available in standard IC form and have been included here as reference.
2-input EX-OR (Exclusive OR) Gate
For a 2-input Ex-OR gate, the output Q is true if
EITHER input A or if input B is true, but NOT both giving the Boolean Expression of:
( Q = (A and NOT B) or (NOT A and B) ).
| Symbol | Truth Table | ||
 | A | B | Q |
| 0 | 0 | 0 | |
| 0 | 1 | 1 | |
| 1 | 0 | 1 | |
| 1 | 1 | 0 | |
| Boolean Expression Q = A⊕B | |||
2-input EX-NOR (Exclusive NOR) Gate
For a 2-input Ex-NOR gate, the output Q is true if BOTH
input A and input B are the same, either true or false, giving the Boolean Expression of:
( Q = (A and B) or (NOT A and NOT B) ).
| Symbol | Truth Table | ||
 | A | B | Q |
| 0 | 0 | 1 | |
| 0 | 1 | 0 | |
| 1 | 0 | 0 | |
| 1 | 1 | 1 | |
| Boolean Expression Q = A ⊕ B | |||
Summary of all the 2-input Gates described above.
The following Truth Table compares the logical functions of the 2-input logic gates above.
| Inputs | Truth Table Outputs for each Gate | ||||||
| A | B | AND | NAND | OR | NOR | EX-OR | EX-NOR |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
The following table gives a list of the common logic functions and their equivalent Boolean notation.
| Logic Function | Boolean Notation |
| AND | A.B |
| OR | A+B |
| NOT | A |
| NAND | A .B |
| NOR | A+B |
| EX-OR | (A.B) + (A.B) or A⊕B |
| EX-NOR | (A.B) + or A ⊕ B |
2-input logic gate truth tables are given here as examples of the operation of each logic function, but
there are many more logic gates with 3, 4 even 8 individual inputs. The multiple input gates are no different to the simple
2-input gates above, So a 4-input AND gate would still require ALL 4-inputs to be present to produce the required output at
Q and its larger truth table would reflect that.
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