Second Semester_Digital Logic_Canonical Expression

 

🔹 Canonical Expression

A canonical expression is a Boolean expression written in a form where each term contains all the variables in the system, either complemented or uncomplemented.

• Two types:

  1. Canonical Sum of Products (SOP) → Each product term (AND) has all the variables.
     Example: For variables $A, B$:

     $$F(A,B) = A'B + AB'$$

     is not canonical, but

     $$F(A,B) = A'B + AB' = A'B + AB' = (A'B') + (A'B) + (AB')$$

     (expanded so each term has all variables) → canonical SOP.

  2. Canonical Product of Sums (POS) → Each sum term (OR) has all the variables.

✅ Key point: Every canonical form corresponds directly to a truth table row (minterm for SOP, maxterm for POS).

which also means;

👉 Every term must contain all variables, either normal or complemented.

✅ Example (2 variables $A, B$):

• Canonical SOP: $F = A'B + AB'$
  (both terms have A and B)

• Canonical POS: $F = (A + B)(A' + B)$
  (each sum has A and B)

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🔹 Standard Form

👉 Just means the expression is written as SOP (Sum of Products) or POS (Product of Sums).

• SOP = OR of AND terms

• POS = AND of OR terms

✅ Example (2 variables $A, B$):

• Standard SOP: $F = A + AB'$

• Standard POS: $F = (A + B)(A' + B)$

⚠️ In standard form, not all variables need to appear in every term.

or We can Tell also as;

A standard form is a Boolean expression written as either:

• Sum of Products (SOP): OR of product terms.

• Product of Sums (POS): AND of sum terms.

⚠️ The difference is:

• In standard SOP/POS, not all variables need to appear in each term.

• In canonical SOP/POS, every term must include all variables.

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✅ Example (3 variables: A, B, C)

• Standard SOP:

  $$F(A,B,C) = AB + BC'$$

  (terms don’t include all variables)

• Canonical SOP:

  $$F(A,B,C) = A'B'C + AB'C' + ABC$$

  (every product has all variables: A, B, and C)

• Standard POS:

  $$F(A,B,C) = (A + B)(B' + C)$$

  (not all variables in each sum)

• Canonical POS:

  $$F(A,B,C) = (A + B + C)(A + B' + C)(A' + B + C')$$

  (every sum includes all variables)

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🌟 In short:

• Standard form = SOP or POS (not necessarily all variables in each term).

• Canonical expression = Special standard form where each term has all variables.