Canonical forms: minterms, maxterms, SOP and POS

0 · Recall Warm-up

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

• Complement of AB + C'? from Lesson 3.1.2
• A + A'B simplifies to? from Lesson 3.1.2
• Rows of the truth table of F(A, B, C) = AB + C' that give 1? from Lesson 3.1.1

1 · The Hook

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Two people may write the same function very differently: A + B'C and AB' + AB + B'C look nothing alike. To compare functions, to give them to K-maps, or to feed them to a computer program, we need one standard way of writing each function. The answer is the list of truth-table rows where it is 1 (or 0). GATE writes functions as F = Σm(…) or ΠM(…) all the time.

2 · Predict First

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F(A, B, C) is 1 exactly in rows 1, 4 and 7. Write one AND term (using every variable once) that is 1 only in row 4 (A B C = 1 0 0).

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

3 · Core Concept

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Predict First — the answer

AB'C': it is 1 only when A = 1, B = 0, C = 0.

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In plain words

For each row where F is 1, write an AND term that is true for that row only (a minterm). OR them together and you have F. Or, for each row where F is 0, write an OR term that is false for that row only (a maxterm), and AND them together. These are the two canonical forms.

1. Minterms

A minterm of n variables is an AND of all n variables, each plain or complemented, that is 1 in exactly one row. Write a variable plain where the row has 1, complemented where it has 0. Its number is the row number. Row 5 of F(A, B, C) is 101 → m₅ = AB'C.

2. Maxterms

A maxterm is an OR of all n variables that is 0 in exactly one row. Write a variable complemented where the row has 1, plain where it has 0. Row 5 (101) → M₅ = A' + B + C'. Note Mᵢ = (mᵢ)'.

3. Canonical SOP and POS

• Canonical sum of products (SOP): F = OR of the minterms of the 1-rows, written F = Σm(…).
• Canonical product of sums (POS): F = AND of the maxterms of the 0-rows, written F = ΠM(…).
• The two lists are complementary: for 3 variables, Σm(1, 4, 5, 6, 7) = ΠM(0, 2, 3).
• F' uses the other rows: if F = Σm(1, 4, 5, 6, 7), then F' = Σm(0, 2, 3).

4. Expanding a shorter expression

A term missing a variable covers several rows: multiply by (X + X') for each missing variable. F(A, B, C) = A + B'C:

• A = A(B + B')(C + C') → rows 4, 5, 6, 7.
• B'C = (A + A')B'C → rows 1, 5.
• F = Σm(1, 4, 5, 6, 7) = ΠM(0, 2, 3).

A product term with k of the n variables covers 2ⁿ⁻ᵏ minterms.

5. Standard (non-canonical) forms

Any OR of AND terms is an SOP (two levels: ANDs then an OR); any AND of OR terms is a POS. Canonical forms are the special case where every term has all variables. SOP and POS are what K-maps (Phase 4) minimise.

4 · Visual

F(A,B,C) = A + B'C       rows with F = 1 -> minterms,  rows with F = 0 -> maxterms

row  A B C  F
 0   0 0 0  0   ->  M0 = (A + B + C)
 1   0 0 1  1   ->  m1 = A'B'C
 2   0 1 0  0   ->  M2 = (A + B' + C)
 3   0 1 1  0   ->  M3 = (A + B' + C')
 4   1 0 0  1   ->  m4
 5   1 0 1  1   ->  m5
 6   1 1 0  1   ->  m6
 7   1 1 1  1   ->  m7
F = Σm(1,4,5,6,7) = ΠM(0,2,3)

5 · Worked Examples

Example 1 Easy

Write minterm m₆ and maxterm M₆ of F(A, B, C).

Solution

Row 110: m₆ = ABC', M₆ = A' + B' + C.

Example 2 Medium

F(A, B, C) = Σm(0, 3, 5). Write F in ΠM notation.

Solution

The other rows: ΠM(1, 2, 4, 6, 7).

Example 3 GATE

How many minterms does the term AC' cover in a function of 4 variables A, B, C, D?

Solution

2 of 4 variables used → 2⁴⁻² = 4 minterms (A = 1, C = 0, B and D free: 8, 9, 12, 13).

6 · GATE Traps

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• Maxterm literals are complemented where the row has 1 (the opposite of minterms).
• ΠM lists the rows where F is 0, not the complement's 1-rows written differently by mistake.
• Σm(F) and ΠM(F) together list every row exactly once.
• The first variable is the MSB when numbering rows.

7 · Memory Hook

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"Minterm: AND, true in one row (1 → plain). Maxterm: OR, false in one row (1 → primed). Σ lists the 1s, Π lists the 0s."

8 · Connect the Dots

Any function can now be built from AND, OR and NOT. Can we do it with fewer kinds of gates? The next lesson shows that NAND alone (or NOR alone) is enough, and how to decide whether any set of gates is enough.

9 · Self-Test

GATE format. Try every question before opening the vault.

Q1MCQ1 markMinterm

For F(A, B, C), minterm m₃ is:

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

Q2MCQ1 markMaxterm

For F(A, B, C), maxterm M₅ is:

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

Q3MCQ2 marksCanonical forms

F(A, B, C) = A + B'C. Which is F in ΠM notation?

• (A) ΠM(4, 5, 6, 7)
• (B) ΠM(0, 1, 2, 3)
• (C) ΠM(1, 4, 5, 6, 7)
• (D) ΠM(0, 2, 3)

Q4NAT1 markCoverage

How many minterms of a 5-variable function does the term BD' cover?

Numerical answer: ________

Q5NAT2 marksComplement

F(A, B, C, D) = Σm(0, 2, 5, 7, 8, 10, 13, 15). How many minterms does F' have?

Numerical answer: ________

10 · Rule Card

• Minterm mᵢ: AND of all variables, plain where row i has 1. Maxterm Mᵢ: OR of all variables, primed where row i has 1. Mᵢ = mᵢ'.
• F = Σm(1-rows) = ΠM(0-rows); F' = Σm(0-rows).
• Expand a term by multiplying with (X + X') for each missing variable; k of n variables → 2ⁿ⁻ᵏ minterms.
• SOP = OR of AND terms; POS = AND of OR terms; canonical = every term has all variables.

11 · Concept Ledger

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

mintermmaxtermminterm numberingcanonical sum of productscanonical product of sumsΣm and ΠM notationcomplement in canonical formexpanding terms to mintermsminterms covered by a termstandard SOP and POS

12 · Answer Vault