Test: POS Form - Electronics and Communication Engineering (ECE) MCQ

• If two functions are represented in terms of max terms, the product will include all the max terms that are present in either of F₁ or F₂. This is because max terms are the values for which the function, gives a 0 output.
• If two functions are represented in terms of minterms, the product will include all the common min terms present in both F₁ and F₂. This is because minterms are the values for which the function gives an output of 1.

Application:
F₁ (a, b, c) = Π M (0, 4, 5, 6)
And F₂ (a, b, c) = Π M (0, 3, 4, 6, 7)
The product F₁F₂  will include all the max terms that are present in either F₁ or F₂
∴ F(a, b, c) = F₁.F₂  = Π M (0, 3, 4, 5, 6, 7)
Alternate Method:
In the form of minterms, the given function can be expressed as:
F₁(a, b, c) = ∑ m (1, 2, 3, 7)
F₂ (a, b, c) = ∑ m (1, 2, 5)
The product F₁F₂ will include minterms that are present in both F₁ and F₂ (common minterms), i.e.
F(a, b, c) = ∑ m (1, 2)
Now, the function can be expressed in the form of max terms as:
F(a, b, c) = ∑ m(1, 2) = Π M (0, 3, 4, 5, 6, 7)

Minterm:
A minterm is a Boolean expression resulting in 1 for the output of a single cell, and 0s for all other cells in a Karnaugh map, or truth table.
0 is represented by A̅  or A'
1 is represented by A
Maxterm:
A maxterm is a Boolean expression resulting in a 0 for the output of a single cell expression, and 1s for all other cells in the Karnaugh map, or truth table.
0 is represented by A 
1 is represented by A̅  or A'
Calculation:
The given Truth Table is:

In POS form we consider only terms with 0 in output.
F = (A + B + C̅)(A̅  + B̅  + C̅)  
Hence option (2) is the correct answer.

Concept:
The SOP representation of the circuit is:
F = Σm (min terms)
The POS representation of the circuit:
F = ΠM (max terms)
Application:

min terms = (0, 2, 4)
max terms = (1, 3, 5, 6, 7)
POS form
F = ΠM (max term)
= ΠM (1, 3, 5, 6, 7)

The K-map is a graphical tool used to simplify a logic equation or to convert a truth table to its corresponding logic circuit in a simple and orderly process.

For 4-variables POS expression, the cell with address 0000 indicates A + B + C + D.
Note: For 4-variables SOP (Sum of Product) expression, the cell with address 0000 will indicate A̅ B̅ C̅ D̅ 

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.
F(x,y,z) = Σm(1,2,5,6,7)
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,
F(x,y,z) = Σm(1,2,5,6,7) = πM(0,3,4)
We get

F = A̅.B̅.C̅ + A̅.B.C̅ + A.B̅.C̅
In terms of minterms, this can be represented as:
F = ∑m (0, 2, 4)
The equivalent maxterm will contain the terms not present in the minterm representation, i.e.
F = ∑m (0, 2, 4) = π(1, 3, 5, 6, 7) = M₁. M₃. M₅. M₆. M₇
⇒ (A + B + C̅) (A + B̅ + C̅) (A̅ + B + C̅) (A̅ + B̅ + C) (A̅ + B̅ + C̅)

F = xy + x’z
F = xy (z+z’) + x’(y+y’)z
= xyz + xyz’ + x’yz + x’y’z
The given expression can be expressed in terms of the sum of minterms as:
= ∑ (1,3,6,7)
Max terms will simply be the missing minterms, i.e. the function in terms of maxterms can be expressed as:
= ∏ (0,2,4,5)
The representation in POS will, therefore, be:
= (x + y + z)( x + y’+ z)(x’ + y + z)(x’ + y + z’)