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Answer the following using a truth table and a Karnaugh map.
A shopping mall announces a special discount on all its products as a festival offer only to those…
A shopping mall announces a special discount on all its products as a festival offer only to those who satisfy any one of the following conditions.
• If he/she is an employee of the mall and has a service of more than 10 years.
OR
• A regular customer of the mall whose age is less than 65 years and should not be an employee of the mall.
OR
• If he/she is a senior citizen but not a regular customer of the mall.
The inputs are :
(In all the above cases, 1 indicates yes and 0 indicates no.)
Output: X - Denotes eligible for discount [1 indicates YES and 0 indicates NO in all cases]
| INPUTS | |
|---|---|
| E | Employee of the mall |
| R | Regular customer of the mall |
| S | Service of the employee is more than 10 years |
| C | Senior citizen of 65 years or above |
(i)[5.0]
Draw the truth table for the inputs and outputs given above and write the SOP expression for $X(E, R, S, C)$.
(ii)(a)[2.5]
Reduce the above expression $X(E, R, S, C)$ by using 4-variable Karnaugh map, showing the various groups (i.e. octal, quads and pairs).
(ii)(b)[2.5]
Draw the logic gate diagram for the reduced expression. Assume that the variables and their complements are available as inputs.
Draw: logic gate diagram for the reduced expression
Answer
Answer
Official answer key3(i): Truth table:
(E,R,S,C 1 = yes, 0 = no.) X = 1 when: (Condition 1) employee with service > 10 years, i.e. E.S; OR (Condition 2) regular customer, not employee, age below 65, i.e. E'.R.C'; OR (Condition 3) senior citizen (C) who is not a regular customer, i.e. C.R'.
$X(E,R,S,C) = \Sigma(1,3,4,6,9,10,11,14,15)$
Canonical SOP:
$X = E'R'S'C + E'R'SC + E'RS'C' + E'RSC' + ER'S'C + ER'SC' + ER'SC + ERSC' + ERSC$
3(ii): 4-variable K-map for $X(E,R,S,C)=\Sigma(1,3,4,6,9,10,11,14,15)$ (rows in order E'R', E'R, ER, ER'; columns in order S'C', S'C, SC, SC' - Gray code order):
Groups formed:
Quad 1: cells (1,3,9,11) -> $R'C$
Quad 2: cells (10,11,14,15) -> $ES$
Pair: cells (4,6) -> $E'RC'$
Reduced SOP expression:
$X(E,R,S,C) = R'C + ES + E'RC'$
| E | R | S | C | X |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 |
From ISC 2025 Specimen Computer Science Paper 1, question 3.
Check your working with the Boolean algebra solver: the steps law by law, the K-map and the logic circuit for any expression.