# How many Boolean expressions can be be formed with 3 Boolean variables?

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How many Boolean expressions can be be formed with 3 Boolean variables?
1. 16
2. 1024
3. 32
4. 256

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Correct Answer - Option 4 : 256

• First, we need to understand that when there are no variables, there are two expressions :
• False=0 and True=1
• For one variable pfour functions can be constructed. A function maps each input value of a variable to one and only one output value.
• The False(pfunction maps each value of p to (False).
• The identity (pfunction maps each value of p to the identical value.
• The flip (pfunction maps False to True and True to False.
• The True (p) function maps each value of p to (True).
• For one variable:
• $2^{2^1}$functions can be constructed.This information can be collected into a table:
•  Input Function p False p -p True 0 0 0 1 1 1 0 1 0 1
• For n Variables:
•  Number of Variables Number of Boolean Functions 0 $2^{2^0}$ = 20 = 2 1 $2^{2^1}$ = 2 = 4 2 $2^{2^2}$ = 24 = 16 3 $2^{2^3}$ = 28 = 256 4 $2^{2^4}$ = 216 = 65536 n $2^{2^n}$
• Therefore, according to the above table, a maximum of 256 Boolean functions can be generated with 3 variables.