Don't-care conditions and 5-variable maps

0 · Recall Warm-up

Answer from memory first. Answers are in the Answer Vault at the bottom.

• What makes a prime implicant essential? from Lesson 4.1.2
• How many minterms does a term with 2 of 4 variables cover? from Lesson 3.2.1
• Which BCD patterns are invalid? from Lesson 1.2.1

1 · The Hook

🎣

A circuit reads a BCD digit (0–9). The inputs 1010 to 1111 can never arrive, so we don't care what the circuit outputs for them. Marking those cells X (don't care) lets us treat them as 1 when that makes a group bigger, often halving the circuit. GATE loves functions given as Σm(…) + d(…). And five-variable maps, built from two four-variable maps, appear now and then.

2 · Predict First

🤔

A BCD input ABCD should give F = 1 when the digit is 5 or more (5, 6, 7, 8, 9). If cells 10–15 may be used freely, can the whole row AB = 10 and row AB = 11 be one group?

Make a guess in your head. The answer opens Section 3.

3 · Core Concept

💡

Predict First — the answer

Yes: rows 11 (12–15) and 10 (8–11) together are 8, 9 plus don't-cares, so the group A is allowed.

🗣️

In plain words

A don't-care cell is a wild card. Use it as a 1 if it lets you draw a bigger group; otherwise leave it out. Never draw a group made only of don't-cares, because it covers nothing that matters.

1. Don't-care conditions

Written F = Σm(…) + d(…), or X in the map. Rules:

• An X may be included in a group as if it were 1 (to make the group bigger).
• An X need not be covered.
• Prime implicants are found with Xs counted as 1s; essential PIs are decided by the real 1s only.

2. Example: BCD digit ≥ 5

F(A, B, C, D) = Σm(5, 6, 7, 8, 9) + d(10, 11, 12, 13, 14, 15).

• A (8–15): uses 8, 9 and six Xs.
• BD (5, 7, 13, 15) and BC (6, 7, 14, 15).
• F = A + BD + BC (5 literals). Without the don't-cares the best is AB'C' + A'BD + A'BC (9 literals).

3. Example with a choice

F = Σm(1, 3, 7, 11, 15) + d(0, 2, 5):

• CD (3, 7, 11, 15) is essential.
• m₁ can go with A'B' (0, 1, 2, 3, using Xs 0, 2) or A'D (1, 3, 5, 7, using X 5).
• Two minimal answers: CD + A'B' or CD + A'D. They differ only on the don't-care inputs.

4. Five-variable maps

Draw two 4-variable maps, one for A = 0 (minterms 0–15) and one for A = 1 (16–31). A cell is also adjacent to the cell in the same position in the other map. A group spanning both maps loses A. F(A, B, C, D, E) = Σm(0, 2, 4, 6, 9, 13, 21, 23, 25, 29, 31):

• A'B'E' (0, 2, 4, 6) in the A = 0 map.
• BD'E (9, 13, 25, 29): same positions in both maps.
• ACE (21, 23, 29, 31) in the A = 1 map.
• F = A'B'E' + BD'E + ACE.

4 · Visual

BCD digit ≥ 5 with don't-cares: F = A + BD + BC.

Left half of a 5-variable map (A = 0).

Right half (A = 1). The orange cells sit in the same positions in both halves, so they form one group BD'E.

5 · Worked Examples

Example 1 Easy

In F = Σm(1, 3) + d(5, 7) with 3 variables, what is the minimal SOP?

Solution

Use the Xs: 1, 3, 5, 7 → C.

Example 2 Medium

F(A, B, C, D) = Σm(1, 3, 7, 11, 15) + d(0, 2, 5). Give one minimal SOP.

Solution

CD + A'B' (or CD + A'D).

Example 3 GATE

F(A, B, C, D) = Σm(1, 3, 5, 7, 9) + d(10, 11, 12, 13, 14, 15) (BCD "digit is odd"). Minimal SOP?

Solution

1, 3, 5, 7, 9 plus Xs 11, 13, 15 fill both D = 1 columns → D.

6 · GATE Traps

⚠️

• Don't-cares are optional: include them only to enlarge groups.
• Never make a group of only don't-cares.
• Essential PIs are judged by the real 1s, not by Xs.
• In 5-variable maps, adjacency across the two halves is by same position, not by touching edges.

7 · Memory Hook

🧠

"X = wild card: use it if it helps, ignore it if it doesn't. Five variables = two maps stacked."

8 · Connect the Dots

Chapter 4.1 is complete. K-maps get hard beyond 5 or 6 variables and depend on spotting groups by eye. The tabular method in Chapter 4.2 finds prime implicants mechanically, for any number of variables.

9 · Self-Test

GATE format. Try every question before opening the vault.

Q1MCQ2 marksDon't-cares

The minimal SOP of F(A, B, C, D) = Σm(5, 6, 7, 8, 9) + d(10, 11, 12, 13, 14, 15) is:

• (A) AB'C' + A'BD + A'BC
• (B) A + B
• (C) A + BD + BC
• (D) A + BC

Q2NAT1 markDon't-cares

F(A, B, C, D) = Σm(1, 3, 5, 7, 9) + d(10, 11, 12, 13, 14, 15). How many literals in the minimal SOP?

Numerical answer: ________

Q3NAT2 marksChoice

F = Σm(1, 3, 7, 11, 15) + d(0, 2, 5). How many different minimal SOPs are there?

Numerical answer: ________

Q4MCQ1 markRules

Which statement about don't-care cells is true?

• (A) A group may consist only of don't-cares
• (B) Every don't-care must be covered
• (C) Don't-cares must be treated as 0
• (D) A don't-care may be treated as 1 to enlarge a group

Q5NAT2 marksFive variables

F(A, B, C, D, E) = Σm(0, 2, 4, 6, 9, 13, 21, 23, 25, 29, 31). How many literals are in its minimal SOP?

Numerical answer: ________

10 · Rule Card

• Don't-cares (d or X): may be used as 1s to enlarge groups; need not be covered; never group only Xs.
• PIs computed with Xs as 1s; essentiality judged on the real 1s.
• 5-variable map: two 4-variable maps (A = 0, A = 1); same position in both halves is adjacent; a group in both halves drops A.

11 · Concept Ledger

Unlocked in this lesson. From here on, these terms may be used.

don't-care conditionsd-notationusing don't-cares in groupsBCD don't-caresmultiple minimal answers with don't-caresfive-variable K-mapsadjacency between the two halves

12 · Answer Vault