K-Map 3 Variable

Three-variable logic functions appear everywhere in control logic: priority encoders, segment decoders, and ALU condition flag generators all reduce to 3-variable K-map problems. This map is the most common format in Indian university examinations.

Core Concept

A 3-variable K-map has 8 cells arranged in a 2×4 grid. The rows are labeled with AB in Gray code order: 00, 01, 11, 10. The columns are labeled with C: 0 then 1. This arrangement means rows AB=01 and AB=11 are adjacent, and rows AB=00 and AB=10 are adjacent — but 00 and 11 are not directly adjacent.

Groups of 1, 2, 4, or 8 cells are allowed. A group of 8 means F=1 always. A group of 4 eliminates 2 variables, leaving 1 literal. A group of 2 eliminates 1 variable, leaving a 2-literal term. The prime implicant is the largest group that cannot be expanded further.

The 74HC32 quad-OR gate at 5 V (t_pd ≈ 7 ns, fan-out 10) implements the OR of simplified terms. The 74HC08 AND provides the product terms. Together they realize any 2-level SOP from a K-map grouping.

Boolean Expression

Read each group: list the variables that are constant across all cells in the group. If a variable is 0 throughout, write it complemented. If 1 throughout, write it uncomplemented. If it changes, omit it. F = Σm(list) specifies the function; the K-map gives the minimal SOP.

Given:
  F(A,B,C) = Σm(1,3,5,6,7)
  (F=1 at minterms 1,3,5,6,7)

Formula / Rule:
  Draw 3-variable K-map, find largest groups of 1-cells

Step by step:
  K-map layout (rows AB=00,01,11,10 × cols C=0,1):
            C=0   C=1
  AB=00:     0     1   ← m0=0, m1=1
  AB=01:     0     1   ← m2=0, m3=1
  AB=11:     1     1   ← m6=1, m7=1
  AB=10:     0     1   ← m4=0, m5=1

  Group 1: Entire C=1 column: m1,m3,m7,m5 — group of 4
    A: 0,0,1,1 → changes → omit
    B: 0,1,1,0 → changes → omit
    C: 1,1,1,1 → constant 1 → keep C
    Term = C

  Group 2: Rows AB=11 and AB=10, both columns: m6,m7,m4,m5 — group of 4
    A: 1,1,1,1 → constant 1 → keep A
    B: 1,1,0,0 → changes → omit
    C: 0,1,0,1 → changes → omit
    Term = A

  All 1-cells covered. F = C + A

Final Answer:
  F = A + C

Exam Tip: The row order 00,01,11,10 is Gray code — it is NOT binary order 00,01,10,11. Placing rows in binary order makes m2 and m3 non-adjacent and m6 and m7 non-adjacent, breaking most groupings. The wrap-around rule also applies vertically: row AB=00 and row AB=10 are adjacent despite being at opposite ends of the map. Always check for wrap-around groups before finalizing the answer.

Key Properties

• 3-variable map: 8 cells, 2 rows × 4 columns, rows in Gray order: 00,01,11,10
• Group of 2 → eliminate 1 variable → 2-literal term; group of 4 → eliminate 2 → 1-literal term
• Wrap-around: top row (AB=00) adjacent to bottom row (AB=10); left column adjacent to right column
• Prime implicant: a group that cannot be merged into a larger group — always prefer larger groups
• Essential prime implicant: the only group covering a specific 1-cell — must be in the final expression
• 74HC32 quad-OR: Vcc 2–6 V, t_pd ≈ 7 ns, fan-out 10 — implements OR of K-map terms
• Don't-care cells (d or X): include in groups when they allow larger groupings; exclude from required coverage

Quick Revision

• 3-variable K-map: 8 cells, Gray code row order is mandatory
• Larger group = simpler term = fewer literals
• A variable is kept in the term only if it is constant (all 0 or all 1) across the group
• Wrap-around holds in both directions — always check edge groups
• Essential prime implicants must be selected first
• Non-essential prime implicants are chosen only to cover remaining uncovered 1-cells
• Exam trap: putting rows in binary order 00,01,10,11 — Gray code 00,01,11,10 is required for adjacency to hold