How Many Different Boolean Functions Can Be Possible With 3 Variables

Counting Boolean role with some variables

Prerequisite – Approved and Standard Grade
In the beneath articles, we will run across some varieties of problems with three variables.

Statement-ane:
Counting the number of Boolean functions possible with two variables such that there are exactly ii min terms.
Explanation:
As we already know that from two variables (a and b) four $(2^2)$ numbers (0, 1, 2, three) can be formed i.eastward, in binary digits 00, 01, 10, 11 and possible Min terms are a'b', a'b, ab', ab respectively which gives '1' as the output for respective binary digits as input.

Thus the number of possible function with 2 variables as input such that there are exactly 2 min terms are,
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Where '4' is the possible number from two variables and 'ii' is the desired number of min terms for which the number of the part needs to be calculated.
Statement-ii:
Counting the number of Boolean functions possible with three variables such that there are exactly three min terms.
Caption:
Every bit we already know that from three variables (a, b and c), 8 $(2^3)$ numbers (0, 1, ii, 3, 4, 5, 6, 7) can exist formed i.due east, in binary digits 000, 001, 010, 011, 100, 101, 110, 111 and possible Min terms are a'b'c', a'b'c, a'bc', a'bc, ab'c, abc', abc respectively which gives 'ane' as the output for respective binary digits as input.

Thus the number of possible role with three variables as input such that there are exactly three min terms are,
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Where '8' is the possible number from iii variables and '3' is the desired number of min terms for which the number of the part need to be calculated.
Statement-3:
Counting the number of boolean functions possible with iii variables such that in that location are atmost four min terms.
Explanation:
Every bit nosotros already know that from three variables (a, b and c), 8 $(2^3)$ numbers (0, 1, ii, iii, iv, 5, half dozen, 7) tin be formed, i.e., in binary digits 000, 001, 010, 011, 100, 101, 110, 111 and possible Min terms are a'b'c', a'b'c, a'bc', a'bc, ab'c, abc', abc respectively which gives '1' as the output for respective binary digits as input.
Thus the number of possible role with three variables every bit input such that in that location are at most 4 min terms are,
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Where '8' is the possible number from three variables and 0, one, 2, 3, four are the desired number of min terms for which the number of the function need to be calculated.
Argument-4:
Counting the number of Boolean functions possible with iii variables such that there are minimum 4 min terms.
Explanation:
As nosotros already know that from iii variables (a, b and c), eight $(2^3)$ numbers (0, 1, 2, iii, 4, 5, half-dozen, 7) can be formed i.e, in binary digits 000, 001, 010, 011, 100, 101, 110, 111 and possible Min terms are a'b'c', a'b'c, a'bc', a'bc, ab'c, abc', abc respectively which gives '1' as the output for corresponding binary digits as input.
Thus the number of possible function with three variables as input such that there are minimum 4 min terms are,
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Where '8' is the possible number from 3 variables and 4, 5, 6, seven are the desired number of min terms for which the number of the function need to be calculated.
Statement-5:
Counting the number of Boolean functions possible with 'chiliad' variables such that at that place are 'm' min terms.
Caption:
As we already know that from 'k' variables, $(2^3)$ numbers can exist formed.
Thus the number of possible function with 'yard' variables as input such that there are 'thousand' min terms are,
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Where $(2^k)$ is the possible number from 'thou' variables and '1000' are the desired number of min terms for which the number of the office need to exist calculated.
Statement-6:
Counting the number of Boolean functions possible in neutral office of 'three' variables where there is equal number of min and max terms.
Caption:
Every bit we already know that from 3 variables (a, b and c), 8 $(2^3)$ numbers (0, ane, two, 3, 4, v, six, 7) tin be formed i.due east, in binary digits 000, 001, 010, 011, 100, 101, 110, 111 and there are equal number of max and min terms which gives 'ane' as the output for min terms and '0' equally the output for max terms for corresponding binary digits as input.

For case, suppose 000 is the binary digits then min terms will be a'b'c' and max terms volition be abc.
Thus the number of possible Boolean office with three variables in the neutral function are,
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Where 'viii' is the possible number from three variables and 4 are the desired number of min terms or max terms for which the number of the function needs to be calculated.
Statement-7:
Counting the number of Boolean functions possible in neutral function of 'k' variables where there is an equal number of min and max terms.
Explanation:
Equally we already know that from 'k' variables, $(2^k)$ numbers tin exist formed.
Thus the number of possible Boolean function with 'chiliad' variables as input where an equal number of min and max terms are,
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Where $(2^k)$ is the possible number from 'k' variables and $(2^{(k-1)})$ are the desired number of either min or max terms for which the number of the function needs to exist calculated.