Understanding Sum of Minterms and Product of Maxterms

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A Boolean function is expressed in two form.

  1. Sum of Minterms
  2. Product of Maxterms

      Sum of Minterms

      x’ y’ z , x y’ z’ , x y’ z, x y z’, x y z  
      gives 1 as output in the above Truth Table.

      Truth Table
      • Literal –  x, y, A, b etc is a label which denote an input variable for a logic gate. Literal can be normal or complimented.
      • Minterm – product of two or more literal using ANDing of each literal.
      • Maxterm – sum of two or more literal using ORing of each literal.

      Before we understand what sum of minterm or product of maxterm is, we must understand a few terminology.

      for example, 

      x or x’, y or y’ 

      for example,

      x.y.z or x’y 

      Suppose we have 2 variable – x and y, then all possible combination of literals are x’y’ , x’y, xy’, xy. If we have 3 variables then all combination of literals are as follows

      Basically, if there are n variable, then there is 2n. For 3 variable, there are 2= 8.

      A minterm is the term from above table that gives 1 output.Let us sum all these terms,

      F = x’ y’ z + x y’ z’  + x y’ z + x y z’ + x y z

         = m1 + m4 + m5 + m6 + m7

      F(x,y,z) = ∑(1,4,5,6,7) is known as Sum of Minterms Canonical Form.

      Why canonical form ?, because all the literals present in each of the terms.

      Product of Maxterm

      The Product of Maxterm is complement of the Sum of Minterm of a function. To obtain the Product of Maxterm, we need two step process.

      1. Find those minterms in the Truth Table that gives a 0 as output.
      2. Complement those minterms using DeMorgan’s law.

      Let us now apply the above to obtain the Product of Maxterm form.
      In the above Truth Table, x’ y’ z’, x’ y z’, x’ y z gives output as 0.

      F = x’ y’ z’ + x’ y z’ + x’ y z                  by Rule 1
         = (x’ y’ z’ + x’ y z’ + x’ y z)’               by DeMorgan’s Law
         = (x + y + z)(x + y’ + z)(x + y’ + z’)
        Product of Maxterms form

      We see that the Product of Maxterm is ANDing of all ORed terms.
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