In a 4-variable K-map, a group of 4 cells all have A=1 and C=0, while B and D take all four combinations (00, 01, 10, 11). The simplified Boolean term for this group is:
AA + C' (sum of the variables that are constant across the group)
BAC' (product of the variables that stay constant — A=1, C=0)
CB'D' (product of the variables that happen to be 0 somewhere in the group)
DABCD' (all four variables included to be safe)
The rule: variables that stay constant across the group form the product term; variables that change are eliminated. Here A=1 (constant, so include as A) and C=0 (constant, so include as C'). B and D each take both values (0 and 1) across the four cells — they vary, so they are eliminated. The result is AC'. Option A is wrong because the simplified term is a product (AND), not a sum (OR). Option D defeats the purpose — including all four variables just restates the unsimplified minterms.
Question 2 Multiple Choice
A K-map has a 1 that can be included in a group of 2 or grouped with more 1s and don't-cares to form a group of 4. The better choice for minimizing the expression is:
AThe group of 2 — smaller groups represent more specific (and therefore more correct) terms
BThe group of 4 — larger groups eliminate more variables and produce a simpler expression
CIt doesn't matter — both groupings produce an equally minimal result
DThe group of 2 — don't-care conditions should be avoided to prevent unintended outputs
A group of 2 eliminates one variable; a group of 4 eliminates two. The goal of K-map simplification is the fewest, largest possible groups — each larger group produces a product term with fewer literals, reducing gate count and complexity. Don't-care conditions (X) are precisely meant to be exploited: they represent inputs that either can't occur or whose output doesn't matter, so you can freely include them to create larger, simpler groups without affecting the circuit's behavior on valid inputs.
Question 3 True / False
In a Karnaugh map, cells in the leftmost column are adjacent to cells in the rightmost column (wrap-around adjacency).
TTrue
FFalse
Answer: True
K-maps use Gray code ordering for both rows and columns, and the map wraps around in both dimensions. The leftmost and rightmost columns are neighbors, as are the top and bottom rows. This wrap-around is essential: without it, some valid Boolean simplifications (where minterms differing in only one variable happen to appear at opposite edges) would be missed. Always check edge and corner cells for grouping opportunities that span the map's boundaries.
Question 4 True / False
Don't-care conditions (marked X) in a K-map should generally be included in a group whenever they are adjacent to a 1.
TTrue
FFalse
Answer: False
Don't-cares may be included in a group if doing so makes the group larger (and therefore the resulting term simpler), but they are never required. An X is an output value you don't care about — you are free to treat it as 0 or 1 depending on which choice is more useful. If including an adjacent X would create a larger valid group, include it. If it doesn't help (or creates a less minimal cover), leave it out. The freedom to choose is the whole point of don't-cares.
Question 5 Short Answer
Why does a larger group in a K-map always correspond to a simpler Boolean expression? What is the relationship between group size and variables eliminated?
Think about your answer, then reveal below.
Model answer: Each doubling of group size eliminates one more variable from the product term. A group of 1 (single cell) produces a 4-literal term (in a 4-variable map). A group of 2 eliminates 1 variable (3-literal term). A group of 4 eliminates 2 variables (2-literal term). A group of 8 eliminates 3 variables (1-literal term). A group of 16 covers the entire map and eliminates all variables (constant 1). This works because the Gray code ordering guarantees that adjacent cells differ in exactly one variable — so cells in a group of 2 share all variables except one, allowing that variable to be removed using A·B + A·B' = A.
The underlying algebraic principle is repeated application of the Boolean identity A·X + A·X' = A: combining two minterms that differ in exactly one variable eliminates that variable. K-maps make this visually obvious by arranging minterms so that spatial adjacency = single-variable difference. Larger groups extend this idea: a group of 4 applies the identity twice, eliminating two variables.