Question

A function of Boolean variables \({\rm{x}},{\rm{y\;and\;z}}\) is expressed in terms of the min-term as  \({\rm{F}}\left( {{\rm{x}},{\rm{y}},{\rm{z}}} \right){\rm{\;}} = {\rm{\;\Sigma \;}}\left( {1,2,5,6,7} \right)\). Which one of the product of sums given below is equal to F(x,y,z)
1. \(\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\)
2. \(\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\)
3. \(\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;\bar z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\)
4. \(\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\)

Answer 1

Correct Answer - Option 2 : \(\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\)

Concept:

A Boolean expression consisting purely of Minterms (product terms) is said to be in the canonical sum of products form.

A Boolean expression consisting purely of Maxterms (sum terms) is said to be in the canonical product of sums form.

a) (A + B)(C + D) – POS form

b) (A)B(C + D) – POS form

c) AB + CD – SOP form

Calculation:

Given logic, expression is minterm expression.

\({\rm{F\;}}\left( {{\rm{x}},{\rm{\;y}},{\rm{\;z}}} \right){\rm{\;}} = {\rm{\Sigma m}}\left( {1,2,5,6,7} \right){\rm{\;}} \)

The max term expression will be formed by the terms which are not present in the min-term expression.

By converting the above min-term expression into max term expression,

\({\rm{F\;}}\left( {{\rm{x}},{\rm{\;y}},{\rm{\;z}}} \right){\rm{\;}} = {\rm{\Sigma m}}\left( {1,2,5,6,7} \right){\rm{\;}} = {\rm{\;\pi M\;}}\left( {0,3,4} \right)\)

We get

\(F= {\rm{\;}}\left( {{\rm{x\;}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\left( {{\rm{x\;}} + {\rm{\;\bar y}} + {\rm{\;\bar z}}} \right)\left( {{\rm{\bar x}} + {\rm{\;y\;}} + {\rm{\;z}}} \right)\)