Truth Tables

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. Logic gate truth tables shows each possible input to the gate or circuit and the resultant output depending upon the combination of the 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 we do not general use ON or OFF but instead give them values of logic level "1" or logic level "0". The four possible combinations for a two-input gate are 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

The output Q is true if both input A, AND input B are both true, (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

2-input OR (Inclusive OR) Gate

The output Q is true if either input A, OR input B is true, (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

The output Q is only true when the input is NOT true, the output is the inverse or complement of the input (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

The output Q is true if both input A and input B are not true, (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 NOT A or NOT B gives Q

2-input NOR (Not OR) Gate

The output Q is true if both input A and input B are not true, (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 NOT A and NOT B gives 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

The output Q is true if either input A or if input B is true, but not both (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

The output Q is true if both input A and input B are the same, either true or false, (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.

The following table gives a list of the common logic functions and their equivalent Boolean notation.