How many boolean functions of $F(x,y,z) = F(x',y,z') + F(x,y',z)$?












2












$begingroup$


How to find total number of boolean functions of the equation :



$F(x,y,z) = F(x',y,z') + F(x,y',z)$





Is there any procedure for this ?










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$endgroup$

















    2












    $begingroup$


    How to find total number of boolean functions of the equation :



    $F(x,y,z) = F(x',y,z') + F(x,y',z)$





    Is there any procedure for this ?










    share|cite|improve this question











    $endgroup$















      2












      2








      2


      4



      $begingroup$


      How to find total number of boolean functions of the equation :



      $F(x,y,z) = F(x',y,z') + F(x,y',z)$





      Is there any procedure for this ?










      share|cite|improve this question











      $endgroup$




      How to find total number of boolean functions of the equation :



      $F(x,y,z) = F(x',y,z') + F(x,y',z)$





      Is there any procedure for this ?







      functions discrete-mathematics boolean-algebra






      share|cite|improve this question















      share|cite|improve this question













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      share|cite|improve this question








      edited Dec 9 '16 at 10:35







      Jon Garrick

















      asked Dec 9 '16 at 9:06









      Jon GarrickJon Garrick

      1,246828




      1,246828






















          1 Answer
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          $begingroup$

          A function with three variables is described in a truth-table with eight (2*2*2) rows:



          x y z F
          0 0 0 f0
          0 0 1 f1
          0 1 0 f2
          0 1 1 f3
          1 0 0 f4
          1 0 1 f5
          1 1 0 f6
          1 1 1 f7


          As any value in the output column can be either 0 or 1, disregarding the equation constraint, the total number of different Boolean functions with three input variables is 2^(2*2*2) = 2^8 = 256



          The given equation translates to a set of equations (+ = or):



          f0 = f5 + f2 =        f2         + f5
          f1 = f4 + f3 = f3 + f4
          f2 = f7 + f0 = f0 + f7
          f3 = f6 + f1 = f1 + f6
          f4 = f1 + f6 = f1 + f6
          f5 = f0 + f7 = f0 + f7
          f6 = f3 + f4 = f3 + f4
          f7 = f2 + f5 = f2 + f5


          Each of the eight equations has one other equation with the same right-hand-side. This leads to:



          f0 = f7
          f1 = f6
          f2 = f5
          f3 = f4


          Inserted in the set of equations:



          f0 = f2 = f5 = f7
          f1 = f3 = f4 = f6


          There are four solutions for (f0; f1): (0; 0), (0; 1), (1; 0) and (1; 1)



          Translated back - as an example - to the original truth-table for (f0=1; f1=0):



          x y z F     x'y z' F1  x y' z F2  F=F1+F2
          0 0 0 f0 1 1 0 1 1 0 1 0 1 1
          0 0 1 f1 0 1 0 0 0 0 1 1 0 0
          0 1 0 f2 1 1 1 1 1 0 0 0 1 1
          0 1 1 f3 0 1 1 0 0 0 0 1 0 0
          1 0 0 f4 0 0 0 1 0 1 1 0 0 0
          1 0 1 f5 1 0 0 0 1 1 1 1 1 1
          1 1 0 f6 0 0 1 1 0 1 0 0 0 0
          1 1 1 f7 1 0 1 0 1 1 0 1 1 1



          The total number of functions which fulfil the equation is four.




          Two of these are trivial constant functions: always 0, or always 1





          Alternative approach using the MiniZinc constraint solver:



           % function F is determined by 2^n = 8 function values:
          array[0..7] of var bool: f;

          function var bool: F(bool: x, bool: y, bool: z) =
          % bool variable are automatically converted (coerced) in 0 .. 1
          f[4*x+2*y+z];

          constraint forall(x,y,z in [false,true]) (
          F(x, y, z) == (F(not x, y, not z) / F(x, not y, z))
          );


          MiniZinc comes up with four solutions as expected:



          [false, false, false, false, false, false, false, false]
          [false, true, false, true, true, false, true, false]
          [true, false, true, false, false, true, false, true ]
          [true, true, true, true, true, true, true, true ]





          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Very Nice Explaination !!
            $endgroup$
            – Jon Garrick
            Dec 9 '16 at 13:07











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          $begingroup$

          A function with three variables is described in a truth-table with eight (2*2*2) rows:



          x y z F
          0 0 0 f0
          0 0 1 f1
          0 1 0 f2
          0 1 1 f3
          1 0 0 f4
          1 0 1 f5
          1 1 0 f6
          1 1 1 f7


          As any value in the output column can be either 0 or 1, disregarding the equation constraint, the total number of different Boolean functions with three input variables is 2^(2*2*2) = 2^8 = 256



          The given equation translates to a set of equations (+ = or):



          f0 = f5 + f2 =        f2         + f5
          f1 = f4 + f3 = f3 + f4
          f2 = f7 + f0 = f0 + f7
          f3 = f6 + f1 = f1 + f6
          f4 = f1 + f6 = f1 + f6
          f5 = f0 + f7 = f0 + f7
          f6 = f3 + f4 = f3 + f4
          f7 = f2 + f5 = f2 + f5


          Each of the eight equations has one other equation with the same right-hand-side. This leads to:



          f0 = f7
          f1 = f6
          f2 = f5
          f3 = f4


          Inserted in the set of equations:



          f0 = f2 = f5 = f7
          f1 = f3 = f4 = f6


          There are four solutions for (f0; f1): (0; 0), (0; 1), (1; 0) and (1; 1)



          Translated back - as an example - to the original truth-table for (f0=1; f1=0):



          x y z F     x'y z' F1  x y' z F2  F=F1+F2
          0 0 0 f0 1 1 0 1 1 0 1 0 1 1
          0 0 1 f1 0 1 0 0 0 0 1 1 0 0
          0 1 0 f2 1 1 1 1 1 0 0 0 1 1
          0 1 1 f3 0 1 1 0 0 0 0 1 0 0
          1 0 0 f4 0 0 0 1 0 1 1 0 0 0
          1 0 1 f5 1 0 0 0 1 1 1 1 1 1
          1 1 0 f6 0 0 1 1 0 1 0 0 0 0
          1 1 1 f7 1 0 1 0 1 1 0 1 1 1



          The total number of functions which fulfil the equation is four.




          Two of these are trivial constant functions: always 0, or always 1





          Alternative approach using the MiniZinc constraint solver:



           % function F is determined by 2^n = 8 function values:
          array[0..7] of var bool: f;

          function var bool: F(bool: x, bool: y, bool: z) =
          % bool variable are automatically converted (coerced) in 0 .. 1
          f[4*x+2*y+z];

          constraint forall(x,y,z in [false,true]) (
          F(x, y, z) == (F(not x, y, not z) / F(x, not y, z))
          );


          MiniZinc comes up with four solutions as expected:



          [false, false, false, false, false, false, false, false]
          [false, true, false, true, true, false, true, false]
          [true, false, true, false, false, true, false, true ]
          [true, true, true, true, true, true, true, true ]





          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Very Nice Explaination !!
            $endgroup$
            – Jon Garrick
            Dec 9 '16 at 13:07
















          2












          $begingroup$

          A function with three variables is described in a truth-table with eight (2*2*2) rows:



          x y z F
          0 0 0 f0
          0 0 1 f1
          0 1 0 f2
          0 1 1 f3
          1 0 0 f4
          1 0 1 f5
          1 1 0 f6
          1 1 1 f7


          As any value in the output column can be either 0 or 1, disregarding the equation constraint, the total number of different Boolean functions with three input variables is 2^(2*2*2) = 2^8 = 256



          The given equation translates to a set of equations (+ = or):



          f0 = f5 + f2 =        f2         + f5
          f1 = f4 + f3 = f3 + f4
          f2 = f7 + f0 = f0 + f7
          f3 = f6 + f1 = f1 + f6
          f4 = f1 + f6 = f1 + f6
          f5 = f0 + f7 = f0 + f7
          f6 = f3 + f4 = f3 + f4
          f7 = f2 + f5 = f2 + f5


          Each of the eight equations has one other equation with the same right-hand-side. This leads to:



          f0 = f7
          f1 = f6
          f2 = f5
          f3 = f4


          Inserted in the set of equations:



          f0 = f2 = f5 = f7
          f1 = f3 = f4 = f6


          There are four solutions for (f0; f1): (0; 0), (0; 1), (1; 0) and (1; 1)



          Translated back - as an example - to the original truth-table for (f0=1; f1=0):



          x y z F     x'y z' F1  x y' z F2  F=F1+F2
          0 0 0 f0 1 1 0 1 1 0 1 0 1 1
          0 0 1 f1 0 1 0 0 0 0 1 1 0 0
          0 1 0 f2 1 1 1 1 1 0 0 0 1 1
          0 1 1 f3 0 1 1 0 0 0 0 1 0 0
          1 0 0 f4 0 0 0 1 0 1 1 0 0 0
          1 0 1 f5 1 0 0 0 1 1 1 1 1 1
          1 1 0 f6 0 0 1 1 0 1 0 0 0 0
          1 1 1 f7 1 0 1 0 1 1 0 1 1 1



          The total number of functions which fulfil the equation is four.




          Two of these are trivial constant functions: always 0, or always 1





          Alternative approach using the MiniZinc constraint solver:



           % function F is determined by 2^n = 8 function values:
          array[0..7] of var bool: f;

          function var bool: F(bool: x, bool: y, bool: z) =
          % bool variable are automatically converted (coerced) in 0 .. 1
          f[4*x+2*y+z];

          constraint forall(x,y,z in [false,true]) (
          F(x, y, z) == (F(not x, y, not z) / F(x, not y, z))
          );


          MiniZinc comes up with four solutions as expected:



          [false, false, false, false, false, false, false, false]
          [false, true, false, true, true, false, true, false]
          [true, false, true, false, false, true, false, true ]
          [true, true, true, true, true, true, true, true ]





          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Very Nice Explaination !!
            $endgroup$
            – Jon Garrick
            Dec 9 '16 at 13:07














          2












          2








          2





          $begingroup$

          A function with three variables is described in a truth-table with eight (2*2*2) rows:



          x y z F
          0 0 0 f0
          0 0 1 f1
          0 1 0 f2
          0 1 1 f3
          1 0 0 f4
          1 0 1 f5
          1 1 0 f6
          1 1 1 f7


          As any value in the output column can be either 0 or 1, disregarding the equation constraint, the total number of different Boolean functions with three input variables is 2^(2*2*2) = 2^8 = 256



          The given equation translates to a set of equations (+ = or):



          f0 = f5 + f2 =        f2         + f5
          f1 = f4 + f3 = f3 + f4
          f2 = f7 + f0 = f0 + f7
          f3 = f6 + f1 = f1 + f6
          f4 = f1 + f6 = f1 + f6
          f5 = f0 + f7 = f0 + f7
          f6 = f3 + f4 = f3 + f4
          f7 = f2 + f5 = f2 + f5


          Each of the eight equations has one other equation with the same right-hand-side. This leads to:



          f0 = f7
          f1 = f6
          f2 = f5
          f3 = f4


          Inserted in the set of equations:



          f0 = f2 = f5 = f7
          f1 = f3 = f4 = f6


          There are four solutions for (f0; f1): (0; 0), (0; 1), (1; 0) and (1; 1)



          Translated back - as an example - to the original truth-table for (f0=1; f1=0):



          x y z F     x'y z' F1  x y' z F2  F=F1+F2
          0 0 0 f0 1 1 0 1 1 0 1 0 1 1
          0 0 1 f1 0 1 0 0 0 0 1 1 0 0
          0 1 0 f2 1 1 1 1 1 0 0 0 1 1
          0 1 1 f3 0 1 1 0 0 0 0 1 0 0
          1 0 0 f4 0 0 0 1 0 1 1 0 0 0
          1 0 1 f5 1 0 0 0 1 1 1 1 1 1
          1 1 0 f6 0 0 1 1 0 1 0 0 0 0
          1 1 1 f7 1 0 1 0 1 1 0 1 1 1



          The total number of functions which fulfil the equation is four.




          Two of these are trivial constant functions: always 0, or always 1





          Alternative approach using the MiniZinc constraint solver:



           % function F is determined by 2^n = 8 function values:
          array[0..7] of var bool: f;

          function var bool: F(bool: x, bool: y, bool: z) =
          % bool variable are automatically converted (coerced) in 0 .. 1
          f[4*x+2*y+z];

          constraint forall(x,y,z in [false,true]) (
          F(x, y, z) == (F(not x, y, not z) / F(x, not y, z))
          );


          MiniZinc comes up with four solutions as expected:



          [false, false, false, false, false, false, false, false]
          [false, true, false, true, true, false, true, false]
          [true, false, true, false, false, true, false, true ]
          [true, true, true, true, true, true, true, true ]





          share|cite|improve this answer











          $endgroup$



          A function with three variables is described in a truth-table with eight (2*2*2) rows:



          x y z F
          0 0 0 f0
          0 0 1 f1
          0 1 0 f2
          0 1 1 f3
          1 0 0 f4
          1 0 1 f5
          1 1 0 f6
          1 1 1 f7


          As any value in the output column can be either 0 or 1, disregarding the equation constraint, the total number of different Boolean functions with three input variables is 2^(2*2*2) = 2^8 = 256



          The given equation translates to a set of equations (+ = or):



          f0 = f5 + f2 =        f2         + f5
          f1 = f4 + f3 = f3 + f4
          f2 = f7 + f0 = f0 + f7
          f3 = f6 + f1 = f1 + f6
          f4 = f1 + f6 = f1 + f6
          f5 = f0 + f7 = f0 + f7
          f6 = f3 + f4 = f3 + f4
          f7 = f2 + f5 = f2 + f5


          Each of the eight equations has one other equation with the same right-hand-side. This leads to:



          f0 = f7
          f1 = f6
          f2 = f5
          f3 = f4


          Inserted in the set of equations:



          f0 = f2 = f5 = f7
          f1 = f3 = f4 = f6


          There are four solutions for (f0; f1): (0; 0), (0; 1), (1; 0) and (1; 1)



          Translated back - as an example - to the original truth-table for (f0=1; f1=0):



          x y z F     x'y z' F1  x y' z F2  F=F1+F2
          0 0 0 f0 1 1 0 1 1 0 1 0 1 1
          0 0 1 f1 0 1 0 0 0 0 1 1 0 0
          0 1 0 f2 1 1 1 1 1 0 0 0 1 1
          0 1 1 f3 0 1 1 0 0 0 0 1 0 0
          1 0 0 f4 0 0 0 1 0 1 1 0 0 0
          1 0 1 f5 1 0 0 0 1 1 1 1 1 1
          1 1 0 f6 0 0 1 1 0 1 0 0 0 0
          1 1 1 f7 1 0 1 0 1 1 0 1 1 1



          The total number of functions which fulfil the equation is four.




          Two of these are trivial constant functions: always 0, or always 1





          Alternative approach using the MiniZinc constraint solver:



           % function F is determined by 2^n = 8 function values:
          array[0..7] of var bool: f;

          function var bool: F(bool: x, bool: y, bool: z) =
          % bool variable are automatically converted (coerced) in 0 .. 1
          f[4*x+2*y+z];

          constraint forall(x,y,z in [false,true]) (
          F(x, y, z) == (F(not x, y, not z) / F(x, not y, z))
          );


          MiniZinc comes up with four solutions as expected:



          [false, false, false, false, false, false, false, false]
          [false, true, false, true, true, false, true, false]
          [true, false, true, false, false, true, false, true ]
          [true, true, true, true, true, true, true, true ]






          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 26 '18 at 12:02

























          answered Dec 9 '16 at 12:20









          Axel KemperAxel Kemper

          3,33611418




          3,33611418












          • $begingroup$
            Very Nice Explaination !!
            $endgroup$
            – Jon Garrick
            Dec 9 '16 at 13:07


















          • $begingroup$
            Very Nice Explaination !!
            $endgroup$
            – Jon Garrick
            Dec 9 '16 at 13:07
















          $begingroup$
          Very Nice Explaination !!
          $endgroup$
          – Jon Garrick
          Dec 9 '16 at 13:07




          $begingroup$
          Very Nice Explaination !!
          $endgroup$
          – Jon Garrick
          Dec 9 '16 at 13:07


















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