Q.1.(b)(i) Convert the following Boolean expression into standard SOP form and also implement the expression.ÂÂ
ABC + AB + ABCD
Q.1.(b)(ii) Convert the following Boolean expression into standard POS form and also implement the expression.
(A + B + C)(B + C + D)(A + B + C + D)
The Correct Answer and Explanation is:
Q.1.(b)(i) Convert to Standard SOP Form and Implement:
Given Boolean Expression:
ABC+AB+ABCDABC + AB + ABCD
Step 1: Eliminate Redundancy
Notice that ABC is already part of AB, and ABCD is more specific than ABC. So the original expression can be simplified: AB+ABCD=AB(1+CD)=AB⋅1=ABAB + ABCD = AB(1 + CD) = AB \cdot 1 = AB
Hence, simplified expression = AB
But we are asked to convert it into Standard SOP (Sum of Products), where each term must include all variables involved (A, B, C, D).
Step 2: Standard SOP Conversion
We express AB in terms of A, B, C, D: AB=AB(C+C′)(D+D′)AB = AB(C + C’)(D + D’)
Use distributive property: =ABCD+ABCD′+ABC′D+ABC′D′= ABCD + ABCD’ + ABC’D + ABC’D’
✅ Standard SOP Form:
ABCD+ABCD′+ABC′D+ABC′D′ABCD + ABCD’ + ABC’D + ABC’D’
Logic Circuit Implementation (SOP):
- Create 4 AND gates for each term:
- Inputs: ABCD, ABCD’, ABC’D, ABC’D’
- OR gate combines the 4 outputs.
Q.1.(b)(ii) Convert to Standard POS Form and Implement:
Given Expression (POS form): (A+B+C)(B+C+D)(A+B+C+D)(A + B + C)(B + C + D)(A + B + C + D)
This is already in Product of Sums (POS) form. But it is not yet in Standard POS, where each sum term must include all variables.
Step 1: Identify All Variables
Variables used: A, B, C, D
Step 2: Expand Each Term to Include All Variables
- (A + B + C) → Add missing D: (A + B + C + D)(A + B + C + D’)
- (B + C + D) → Add missing A: (A + B + C + D)(A’ + B + C + D)
- (A + B + C + D) already includes all variables.
Standard POS Form:
(A+B+C+D)(A+B+C+D′)(A′+B+C+D)(A + B + C + D)(A + B + C + D’)(A’ + B + C + D)
Logic Circuit Implementation (POS):
- 3 OR gates for each clause:
- Inputs: (A + B + C + D), (A + B + C + D’), (A’ + B + C + D)
- Feed all OR outputs into an AND gate.
✅ Summary Explanation
Boolean algebra expressions often require standardization into canonical forms for systematic simplification and implementation in digital circuits. The two primary canonical forms are Sum of Products (SOP) and Product of Sums (POS).
In Q.1.(b)(i), we started with the expression ABC + AB + ABCD. Noticing redundancy, we simplified it to AB, as it encompasses all original terms. However, SOP form requires each product term to have all variables involved—in this case, A, B, C, and D. We expanded AB using identities such as X = X(Y + Y') to generate all possible combinations of the missing variables, resulting in: ABCD+ABCD′+ABC′D+ABC′D′ABCD + ABCD’ + ABC’D + ABC’D’
This SOP form is then implemented using AND gates for each term and a final OR gate to sum the outputs.
For Q.1.(b)(ii), the given expression was already in POS format but not standard. A standard POS form mandates that every sum term includes all variables. We augmented each clause by adding missing variables with both complements and original forms using the identity X = (X + Y)(X + Y'). The result was: (A+B+C+D)(A+B+C+D′)(A′+B+C+D)(A + B + C + D)(A + B + C + D’)(A’ + B + C + D)
This form ensures complete coverage of input combinations. It can be implemented using OR gates per clause and an AND gate to combine them, reflecting the POS structure. These standard forms are crucial for minimizing logic and ensuring consistent digital design.
