# Boolean Algebra 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 standard table to give a visual representation of the switching function of the system.

The table used to represent the boolean expression of a logic gate function 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”.

So in order to keep things simple to understand, in this tutorial we will only deal with standard 2-input type logic gates, but the principals are still the same for gates with more than two inputs.

Then the Truth tables for a 2-input AND Gate, a 2-input OR Gate and a single input 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 (Inverter)

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 inversion of A gives Q

The NAND and the NOR Gates are a combination of the AND and OR Gates respectively with that of a NOT Gate (inverter).

## 2-input NAND (Not AND) Gate

For a 2-input NAND gate, the output Q is NOT true if BOTH input A and input B are 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 Boolean expression to indicate an Exclusive-OR or Exclusive-NOR function is to a symbol with a plus sign inside a circle, (  ).

The switching actions of both of these types of gates can be created using the above standard logic 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 2-input Logic Gates

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) + (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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