Karnaugh Map (K-Map) MCQs

Correct Answer: b) 8

Explanation:
The number of cells in a K-map equals the number of possible input combinations: - For n variables: 2ⁿ cells - 3 variables: 2³ = 8 cells The 3-variable K-map is typically arranged as: - 2 rows (A variable) - 4 columns (BC variables in Gray code order: 00, 01, 11, 10)

Correct Answer: c) Groups of 3

Explanation:
K-map grouping rules: - Groups must contain 2ⁿ cells (1, 2, 4, 8, 16, etc.) - Groups of 3 are invalid because: * They don't eliminate a variable in simplification * They don't represent valid product terms Valid groups always cover adjacent cells where one variable changes state

Correct Answer: a) As large as possible

Explanation:
K-map minimization principles: - Larger groups eliminate more variables - Each group must be as large as possible to: * Minimize the number of product terms * Minimize the literals in each term - The optimal solution uses: * Minimum number of largest possible groups * All 1s must be covered (for SOP)

Correct Answer: c) 16

Explanation:
In a 4-variable K-map (16 cells): - Maximum possible group covers all cells (16) - This represents the constant function F = 1 - Other possible group sizes: 1, 2, 4, 8 - A group of 16 means: * All 4 variables are eliminated * The function is always true regardless of input

Correct Answer: a) 1

Explanation:
When all cells in a K-map are 1: - The entire map can be grouped as one large group - For 3 variables (8 cells), this group covers all minterms - The simplified function is the constant 1 - This means: * The output is always true * No variables appear in the simplified expression * All possible input combinations produce output 1

Correct Answer: d) Horizontally and vertically adjacent cells

Explanation:
K-map adjacency rules: - Cells can be grouped horizontally or vertically - The map "wraps around" (top-bottom and left-right edges are adjacent) - Diagonal cells cannot be combined (they differ in more than one variable) - 4-variable K-map adjacency includes: * Immediate left/right neighbors * Immediate top/bottom neighbors * Cells in the same column at top and bottom * Cells in the same row at left and right

Correct Answer: b) 2

Explanation:
Analyzing the minterms: 1. Plot on 4-variable K-map: - m0-m3: A'B' (top-left corner) - m8-m11: AB' (top-right corner) 2. Possible 4-cell groups: - Group 1: m0-m3 (A'B') - Group 2: m8-m11 (AB') 3. Each group covers 4 adjacent cells 4. These are the only possible 4-cell groups for this function The simplified expression would be: A'B' + AB' = B'

Correct Answer: b) A ⊕ B ⊕ C

Explanation:
Analysis of the pattern: 1. The minterms correspond to all odd-numbered cells 2. In binary, these have LSB (D) = 1 3. The pattern represents the XOR of all variables: - XOR function gives 1 for odd number of 1s in inputs - For 4 variables: A ⊕ B ⊕ C ⊕ D 4. However, the given options suggest it's a 3-variable function (missing D) 5. The correct simplified expression is actually D (all minterms have D=1) There appears to be an error in the question options.

Correct Answer: a) A'B + BC

Explanation:
K-map solution steps: 1. Plot minterms on 3-variable K-map: - m1: A'B'C - m3: A'BC - m4: AB'C' - m6: ABC' 2. Form groups: - Group m1+m3: A'B (eliminates C) - Group m4+m6: BC' (eliminates A) 3. The simplified expression is the sum of these prime implicants: A'B + BC' Note: There appears to be a discrepancy with the options - BC' is the correct second term, not BC.

Correct Answer: d) 8

Explanation:
Essential prime implicants (EPIs): - An EPI covers at least one minterm not covered by any other implicant - In a 4-variable K-map (16 cells), the maximum occurs when: * Each EPI covers exactly 2 cells * No overlapping coverage * 8 groups of 2 cells each (8 EPIs) - This would require a very specific pattern where: * Each 1 in the map is adjacent to exactly one other 1 * No larger groups are possible - In practice, most functions have fewer EPIs