Question

GivenF1 M(0,4,5,6) and F2 M(0,4,7),find the maxterm expansion for F1F2. State a general rule for finding the maxterm expansion of F1F2 given the maxterm expansions of F1 and F2. Prove your answer by using the general form of the maxterm expansion.

Answer 1

Given that $F_1 = \Pi\Mu(0,4,5,6)$ and $F_2 = \Pi\Mu(0,4,7)$ .

Let there be three literals A, B and C, truth table for the functions F₁ , F₂ and F₁ F_{2 ,} is created below

From truth table we can write the Maxterm expression for F₁ F₂

$$F_1F_2 = \Pi \Mu(0,4,5,6,7)$$

A General rule therefore can be formulated for F₁ F₂ as,

The Maxterm expression of product F₁ F₂ contains the Maxterms present in either F₁ or F₂ .

It can be proved using general maxterm expansion.

$$F_1 = \displaystyle\prod_{i=0}^7 (a_i + M_i)$$

$$F_2 = \displaystyle\prod_{j=0}^7 (b_j + M_j)$$

$$F_1F_2 = \displaystyle\prod_{i,j=0}^7 (a_i+b_j + M_i + M_j)$$

the index for P.O.S is varying from 0 to 7, which is same for both indices i and j, and M_i = M_j , therefore it can be written as

$$F_1F_2 = \displaystyle\prod_{i=0}^7 (a_i+b_i + M_i)$$

Which shows that the product of F₁F₂ will have terms of both F₁ and F₂ , with common terms written once.

The Maxterm expression of product F₁F₂ contains the Maxterms present in either F₁ or F₂. It can be proved using general maxterm expansion.