Karnaugh Maps (K-Maps) for ISC Class 12: SOP and POS Reduction Step by Step

Karnaugh map questions carry some of the easiest marks in the ISC Computer Science theory paper, as long as you follow the method carefully. This guide covers everything you need: how to lay out the map, how to group, and how to read off SOP and POS expressions, with fully solved examples.

Why K-maps?

A Karnaugh map is a grid that places the outputs of a Boolean function so that neighbouring cells differ in exactly one variable. Grouping neighbouring 1s (or 0s) removes that variable, which gives the simplest expression without long algebra.

Cell layout: Gray code order

Rows and columns follow Gray code order: 00, 01, 11, 10, not 00, 01, 10, 11. Only one bit changes between neighbours.

3-variable map, F(A, B, C)

            B'C'   B'C   BC    BC'
            (00)   (01)  (11)  (10)
  A' (0)  |  m0  |  m1  |  m3  |  m2  |
  A  (1)  |  m4  |  m5  |  m7  |  m6  |

4-variable map, F(A, B, C, D)

            C'D'   C'D   CD    CD'
  A'B'    |  m0  |  m1  |  m3  |  m2  |
  A'B     |  m4  |  m5  |  m7  |  m6  |
  AB      | m12  | m13  | m15  | m14  |
  AB'     |  m8  |  m9  | m11  | m10  |

Memorise this numbering. Many students lose marks by putting m2 and m3 in the wrong columns.

Grouping rules

1. Group only adjacent cells: horizontally or vertically, never diagonally.
2. A group must contain 1, 2, 4, 8 or 16 cells (a power of 2).
3. Make each group as large as possible.
4. Use the fewest groups that cover every 1.
5. Groups may overlap.
6. The map wraps around: the left edge is adjacent to the right edge, the top to the bottom, and the four corners form one group.

Reading a group: a variable that stays the same throughout the group is kept (complemented if it is 0). A variable that changes is eliminated. A pair removes one variable, a quad removes two, and an octet removes three.

Solved example 1: 3-variable SOP

Reduce F(A, B, C) = Σ(0, 2, 3, 4, 6).

            B'C'  B'C   BC   BC'
  A'      |  1  |  0  |  1  |  1  |
  A       |  1  |  0  |  0  |  1  |

• Quad (m0, m2, m4, m6): columns B'C' and BC' wrap around. C = 0 in all four cells, so the term is C'.
• Pair (m2, m3): A = 0 and B = 1, so the term is A'B.

F = C' + A'B

Solved example 2: 4-variable SOP

Reduce F(A, B, C, D) = Σ(0, 1, 4, 5, 10, 11, 14, 15).

• Quad (m0, m1, m4, m5): A = 0 and C = 0 throughout, so the term is A'C'.
• Quad (m10, m11, m14, m15): A = 1 and C = 1 throughout, so the term is AC.

F = A'C' + AC, which is the XNOR of A and C (A ⊙ C).

Solved example 3: POS reduction

For POS, group the 0s instead. In each group, a variable that is 0 is written as it is, and a variable that is 1 is complemented. The terms are then joined with AND.

Reduce F(A, B, C) = π(1, 3, 5, 6, 7).

            B'C'  B'C   BC   BC'
  A'      |  1  |  0  |  0  |  1  |
  A       |  1  |  0  |  0  |  0  |

• Quad of 0s (m1, m3, m5, m7): C = 1 throughout, so the term is (C').
• Pair of 0s (m6, m7): A = 1 and B = 1, so the term is (A' + B').

F = C' · (A' + B')

Check: m6 = 110 gives A' + B' = 0, so F = 0, which is correct. m0 = 000 gives C' = 1 and A' + B' = 1, so F = 1, which is also correct.

Converting between Σ and π

The minterms that are not listed in Σ are exactly the maxterms in π. For a 3-variable function:

Σ(0, 2, 4, 6) = π(1, 3, 5, 7)

Mistakes that cost marks

• Writing the columns in the order 00, 01, 10, 11 instead of the Gray code order 00, 01, 11, 10.
• Missing wrap-around groups, especially the four corners in a 4-variable map.
• Making a group of 3 or 6 cells. Group sizes must be powers of 2.
• Using SOP reading rules on a POS map. In POS, 0 means the variable as it is, and 1 means the complement.
• Not showing the map. In the exam, always draw and label the K-map and each group, because step marks are awarded.

Practice

1. Reduce F(A, B, C, D) = Σ(0, 2, 8, 10).
2. Reduce F(P, Q, R) = π(0, 2, 4, 6).
3. Reduce F(A, B, C) = Σ(1, 3, 5, 7).

Answers:

1. B'D': the four corners.
2. R: all the 0s have R = 0, which in POS gives the term (R).
3. C

Then try the full Boolean algebra questions in our solved ISC Class 12 Computer Science Sample Paper 2027.