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Quiz: Minterm & Maxterm Expansions

Test your understanding of canonical forms, minterm and maxterm representations, notation, and Shannon expansion with these questions.

Question 1

What distinguishes a canonical form from a standard form in Boolean algebra?

  1. Canonical forms use only NAND gates
  2. In canonical form, every variable appears exactly once (complemented or uncomplemented) in every term
  3. Standard forms are always longer than canonical forms
  4. Canonical forms cannot represent all Boolean functions
Show Answer

Correct Answer: B

In a canonical form, every variable appears exactly once in every term — either complemented or uncomplemented. This makes canonical forms unique representations of Boolean functions: a given function has exactly one canonical SOP (sum of minterms) and one canonical POS (product of maxterms). Standard forms may have variables missing from some terms, so the same function can have multiple different standard SOP or POS expressions.

Concept Tested: Canonical Form

Question 2

For the 3-variable input combination \(ABC = 101\), what is the corresponding minterm expression and its index?

  1. \(\overline{A}B\overline{C}\), designated \(m_2\)
  2. \(A + \overline{B} + C\), designated \(M_5\)
  3. \(A\overline{B}C\), designated \(m_3\)
  4. \(A\overline{B}C\), designated \(m_5\)
Show Answer

Correct Answer: D

A minterm is an AND of all variables, each appearing uncomplemented if its value is 1 and complemented if its value is 0. For \(ABC = 101\): \(A = 1\) (include \(A\)), \(B = 0\) (include \(\overline{B}\)), \(C = 1\) (include \(C\)). The minterm is \(A\overline{B}C\). Its index is the decimal equivalent of the binary input: \(101_2 = 5_{10}\), so it is designated \(m_5\). Option C has the correct expression but wrong index.

Concept Tested: Minterm Construction and Designation

Question 3

What is the fundamental relationship between a minterm \(m_i\) and the maxterm \(M_i\) with the same index?

  1. They are complements of each other: \(m_i = \overline{M_i}\)
  2. They are identical expressions
  3. \(M_i\) has twice as many literals as \(m_i\)
  4. They cover the same set of input combinations
Show Answer

Correct Answer: A

A minterm and maxterm with the same index are complements: \(m_i = \overline{M_i}\) and \(M_i = \overline{m_i}\). For example, \(m_5 = A\overline{B}C\) and \(M_5 = (\overline{A} + B + \overline{C})\). Applying DeMorgan's theorem to \(\overline{m_5}\): \(\overline{A\overline{B}C} = \overline{A} + B + \overline{C} = M_5\). Minterm \(m_i\) equals 1 for exactly one input combination; maxterm \(M_i\) equals 0 for that same combination.

Concept Tested: Minterm-to-Maxterm Relationship

Question 4

What does the notation \(F(A,B,C) = \Sigma m(1,3,5)\) represent?

  1. \(F\) equals the product of minterms 1, 3, and 5
  2. \(F\) equals the product of maxterms 1, 3, and 5
  3. \(F\) equals the sum (OR) of minterms \(m_1\), \(m_3\), and \(m_5\)
  4. \(F\) is undefined for inputs 1, 3, and 5
Show Answer

Correct Answer: C

The sigma notation \(\Sigma m\) represents the sum (OR) of minterms. \(F = \Sigma m(1,3,5) = m_1 + m_3 + m_5 = \overline{A}\overline{B}C + \overline{A}BC + A\overline{B}C\). This is the canonical Sum of Products (SOP) form. The function \(F = 1\) for exactly the input combinations whose decimal indices are listed: inputs 001, 011, and 101.

Concept Tested: Sigma (Sum of Minterms) Notation

Question 5

How do you convert \(F = \Sigma m(1,3,5)\) to \(\Pi M\) notation for 3 variables?

  1. Use the same indices: \(\Pi M(1,3,5)\)
  2. Use the complementary indices: \(\Pi M(0,2,4,6,7)\)
  3. Use all indices: \(\Pi M(0,1,2,3,4,5,6,7)\)
  4. Multiply each index by 2
Show Answer

Correct Answer: B

To convert from \(\Sigma m\) to \(\Pi M\), use all indices NOT in the minterm list. For 3 variables, indices range from 0 to \(2^3 - 1 = 7\). If the ON-set (minterms) is \(\{1,3,5\}\), the OFF-set (maxterms) is \(\{0,2,4,6,7\}\). Therefore \(F = \Sigma m(1,3,5) = \Pi M(0,2,4,6,7)\).

Concept Tested: Converting SOP to POS / Complementary Index Sets

Question 6

For a 3-variable function \(F = \Sigma m(2,4,7)\), what is the complement \(\overline{F}\) expressed in both \(\Sigma m\) and \(\Pi M\) notation?

  1. \(\overline{F} = \Sigma m(2,4,7)\) only
  2. \(\overline{F} = \Pi M(2,4,7)\) only
  3. \(\overline{F} = \Sigma m(0,1,3,5,6)\) only
  4. \(\overline{F} = \Sigma m(0,1,3,5,6) = \Pi M(2,4,7)\) — both are valid
Show Answer

Correct Answer: D

The complement \(\overline{F}\) has its ON-set where \(F\) has its OFF-set and vice versa. So \(\overline{F} = \Sigma m(0,1,3,5,6)\) (the complementary minterm indices). Equivalently, \(\overline{F} = \Pi M(2,4,7)\) (same indices as the original \(F\)'s minterms, but using maxterms). Both representations are valid and equivalent.

Concept Tested: Complement of a Boolean Function

Question 7

In the Shannon expansion \(F = X \cdot F_X + \overline{X} \cdot F_{\overline{X}}\), what is \(F_X\) called and how is it computed?

  1. The positive cofactor — \(F\) evaluated with \(X\) set to 1
  2. The Shannon remainder — the portion of \(F\) independent of \(X\)
  3. The X-factor — the derivative of \(F\) with respect to \(X\)
  4. The residue — the terms containing \(X\)
Show Answer

Correct Answer: A

\(F_X\) is called the positive cofactor of \(F\) with respect to \(X\). It is obtained by setting \(X = 1\) in the expression for \(F\) and simplifying. Similarly, \(F_{\overline{X}}\) is the negative cofactor, obtained by setting \(X = 0\). Shannon expansion decomposes any Boolean function into two subfunctions using these cofactors.

Concept Tested: Cofactor / Shannon Expansion Theorem

Question 8

What are the three sets that partition all \(2^n\) minterms for an incompletely specified function?

  1. Input-set, Output-set, Control-set
  2. On-set, Off-set, DC-set (don't care set)
  3. True-set, False-set, Maybe-set
  4. On-set, Off-set, Error-set
Show Answer

Correct Answer: B

An incompletely specified function partitions all \(2^n\) possible input combinations into three mutually exclusive sets: the On-set (where \(F = 1\)), the Off-set (where \(F = 0\)), and the DC-set (don't care conditions, where \(F\) can be assigned either 0 or 1 during optimization). During minimization, DC-set minterms can be included in or excluded from prime implicant groups to achieve simpler expressions.

Concept Tested: On-Set, Off-Set, DC-Set (Incompletely Specified Functions)

Question 9

Why is literal count an important metric when comparing different Boolean expressions for the same function?

  1. It determines the number of rows in the truth table
  2. It indicates how many input variables the function has
  3. It correlates directly with gate input count and implementation cost
  4. It shows the number of minterms in the function's ON-set
Show Answer

Correct Answer: C

Literal count measures expression complexity — each literal (a variable or its complement) corresponds to one gate input in the hardware implementation. More literals generally means more gate inputs, more wiring, and higher area and power cost. For example, \(F = AB + \overline{A}C\) has 4 literals while \(F = AB + \overline{A}C + BC\) has 6 literals for the same function, making the first expression cheaper to implement.

Concept Tested: Literal Count / Expression Cost

Question 10

In maxterm construction, when an input variable has value 1, how does it appear in the maxterm?

  1. Uncomplemented (e.g., \(A\))
  2. Omitted from the maxterm entirely
  3. Doubled for emphasis (e.g., \(A \cdot A\))
  4. Complemented (e.g., \(\overline{A}\))
Show Answer

Correct Answer: D

Maxterm construction is the dual of minterm construction: variables appear complemented when their value is 1, and uncomplemented when their value is 0. This ensures the maxterm evaluates to 0 for its designated input combination. For \(ABC = 110\): \(M_6 = (\overline{A} + \overline{B} + C)\). Checking: \(\overline{1} + \overline{1} + 0 = 0 + 0 + 0 = 0\), confirming \(M_6 = 0\) for input 110.

Concept Tested: Maxterm Construction

Answers Summary

Question Answer Concept
1 B Canonical Form
2 D Minterm Construction and Designation
3 A Minterm-to-Maxterm Relationship
4 C Sigma Notation (Sum of Minterms)
5 B Converting SOP to POS
6 D Complement of a Boolean Function
7 A Cofactor / Shannon Expansion
8 B On-Set, Off-Set, DC-Set
9 C Literal Count / Expression Cost
10 D Maxterm Construction