Boolean Theorems Expressing Logic Circuits Analysis of Circuits
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Chapter 5 Boolean Algebra SKEE 1223 Digital Electronics Mun’im/Arif/Izam {munim, arif, e-izam}@utm. my FKE, Universiti Teknologi Malaysia January 28, 2017 Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Overview 1 Boolean Theorems 2 Expressing Logic Circuits 3 Analysis of Circuits 4 Canonical Forms 5 Boolean Simplification Canonical Forms Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Boolean Axioms Axiom (or postulate): mathematical statement without proof Axiom 1: A = 0 if A l= 1 A = 1 if A l= 0 Axiom 2: Axiom 3: 0 · 0 = 0 1 · 1 = 1 0 · 1 = 1 · 0 = 0 1 + 1 =1 0 + 0 =0 0+1=1+0=1
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Single Variable Theorems Theorem AND Form OR Form Annulment A· 0 = 0 A+ 1 = 1 Identity A· 1 = A A+ 0 = A Idempotency A·A = A A+ A = A Complementation A·A = 0 A+ A = 1 Involution A=A Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Annulment Theorem A 0 0 A 1 1 A + 1 =1 A • 0=0 A 0 0 1 1 B 0 1 F 0 0 0 1 A 0 0 1 1 B 0 1 F 0 1 1 1
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Identity Theorem A A 1 A A 0 A • 1=A A 0 0 1 1 B 0 1 F 0 0 0 1 A + 0 =A A 0 0 1 1 B 0 1 F 0 1 1 1
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Idempotency Theorem A A A • A=A A 0 0 1 1 B 0 1 F 0 0 0 1 A + A =A A 0 0 1 1 B 0 1 F 0 1 1 1
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Complements Theorem A 0 0 1 1 B 0 1 F 0 0 0 1 A • A=0 A 1 A 0 0 1 1 B 0 1 F 0 1 1 1 A + A =1
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Multivariable Theorems Theorem Function Dual Commutation AB = BA A+ B = B + A Association (AB)C = A(BC) = ABC (A + B ) + C = A + ( B + C) = A+ B + C Distribution A(B + C) = AB + AC A + BC = (A + B)(A + C) De Morgan’s A + B = A·B = A + B
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Multivariable Theorems: Commutation A B B A AB AB A B B A A+B Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Multivariable Theorems: Association A B (AB)C C A B C ABC A B C A(BC)
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Multivariable Theorems: Association A B (A+B)+C C A B C A+B+C A B C A+(B+C)
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Theorems Theorem Function Dual Uniting AB + AB = A (A + B) = A Absorption A + AB = A A(A + B) = A Adsorption A + AB = A + B (A + B)B = AB Consensus AB + AC = AB + BC + AC (A + B)(A + C) = (A + B)(B + C)(A + C) Multiplying (A + B)(A + C) = AC + AB AB + AC = (A + C)(A + B) De Morgan’s A + B = A·B = A + B
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms De Morgan’s Theorem Useful for simplifying inverted variables: A·B = A + B = A·B break! AB A +B A B AB Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Proofs Prove that A + AB = A. Prove that A· (A + B) = A. Prove that A + AB = A + B. Prove that (A + B)(A + C) = A + BC. Canonical Forms Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms of Expression Basic Rules: Connected gates form a logic network An output can be connected to 1 or more inputs An input can only be connected to 1 output. Most common is AND-OR network Two forms of expression Sum-of-Products (SOP) Product-of-Sums (POS). Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms POS to SOP Easy Just multiply through and simplify. Example: Convert (A + C)(B + C¯) to its SOP equivalent. F = (A + C)(B + C¯) = AB + AC¯+ BC + CC¯ = AB + AC¯+ BC + 0 = AB + AC¯+ BC Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification SOP to POS Apply De. Morgan’s twice. Example: Convert F = AB¯+ BC¯to its POS equivalent. 1 Apply De. Morgan’s law once to get the function’s inverse: F = AB¯+ BC¯ = AB¯·BC¯ = (A¯+ B)(B¯ + C) = A¯B¯+ A¯C+ BB¯+ BC = A¯B¯+ A¯C+ BC
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms SOP to POS 2 Apply De. Morgan’s law again: F = F¯ = A¯B¯+ A¯C+ BC = A¯B¯· A¯C·BC = (A + B)(A + C¯)(B¯+ C¯) Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Describing Logic Circuits Algebraically Rules to build an AND-OR network: 2 Each AND operation requires one AND gate Each OR operation requires one OR gate 3 Each complemented variable requires a NOT gate. 1 Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Describing Logic Circuits Algebraically A B B A ABC F = AB¯C +A¯B +ABC +A¯C AB Analysis: F C ABC 1 3 variables AC 2 4 product terms 3 10 literals
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Analysis of Circuits Steps for analyzing a circuit: 2 The inputs to all gates must have a variable name. Write the expression performed by each gate at its output. Do it from left to right. 3 The output of the last gate is the network equation 1 A B 1 A 2 AB 4 C 3 F = AB + AC AC Logic circuit the expression F = AB + AC.
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Analysis of Circuits Truth table for F = AB + BC A 0 0 1 1 B 0 0 1 1 C 0 1 0 1 A 1 1 0 0 AB 0 0 1 1 0 0 AC 0 0 0 1 F 0 0 1 1 0 1 Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms What & Why Canonical Forms Canonical form = standard form Useful for comparing seemingly different functions Unique for each function Two versions: SOP POS Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Minterms & Maxterms Must know Minterms & Maxterms to get canonical form. Minterm: AND term with every variable present in either true or complemented form. Maxterm: OR term with every variable present in either true or complemented form.
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Minterms for 2 -variable Function A B Minterm Maxterm 0 0 1 1 0 1 m 0 = A¯B¯ M 0 = A + B m 1 = A¯B M 1 = A + B¯ m 2 = AB¯ M 2 = A¯+ B m 3 = AB M 3 = A¯+ B¯ Important: In a minterm or maxterm, each variable appears once, either as the variable itself or as the inverse For minterms: 1 means the variable is not complemented and 0 means the variable is complemented. For maxterms: 0 means the variable is not complemented and 1 means the variable is complemented.
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Minterms for 3 -variable Function A B C Minterm Maxterm 0 0 1 1 0 1 0 1 m 0 = A¯B¯C¯ m 1 = A¯B¯C m 2 = A¯BC¯ m 3 = A¯BC m 4 = AB¯C¯ m 5 = AB¯C m 6 = ABC¯ m 7 = ABC M 0 = A + B + C M 1 = A + B + C¯ M 2 = A + B¯+ C M 3 = A + B¯+ C¯ M 4 = A¯+ B + C M 5 = A¯+ B + C¯ M 6 = A¯+ B¯+ C M 7 = A¯+ B¯+ C¯ Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Canonical SOP A 0 0 1 1 B 0 0 1 1 C 0 1 0 1 F 1 0 0 0 1 1 → A¯B¯C¯→ m 0 ¯ m 2 → A¯BC → → ABC¯→ m 6 → ABC → m 7 ∴ F (A, B, C) = A¯B¯C¯+ A¯BC¯+ ABC = m 0 + m 2 + m 6 + m 7 = ∑ m(0, 2, 6, 7) Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Canonical From Non-Canonical SOP Expand terms using A + A¯= 1. F = AB(C + C¯) + AC¯(B + B¯) + BC(A + A¯) = ABC + ABC¯+ AB¯C¯+ ABC + A¯BC = A¯BC + AB¯C¯+ ABC = m 3 + m 4 + m 6 + m 7 = ∑ m(3, 4, 6, 7) Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Canonical POS A 0 0 1 1 B 0 0 1 1 C 0 1 0 1 F 1 0 0 0 1 1 → A + B + C¯→ M 1 → A + B¯+ C¯→ M 3 → A¯+ B + C → M 4 → A¯+ B + C¯→ M 5 ∴ F (A, B, C) = (A + B + C¯)(A + B¯+ C¯)(A¯+ B + C)(A¯ + B + C¯) = M 1 M 3 M 4 M 5 = ∏ M(1, 3, 4, 5)
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Simplification Procedure 1 Convert to SOP (sum of products) form. (A + B)(C + D) → AC + AD + BC + BD 2 Check for exact duplicates and drop. AB + BC + AB → AB + BC 3 Check for null values and drop. A(A + B) → AA + AB → AB Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Canonical Forms Boolean Simplification Procedure 4 See if one of the items is entirely contained in another. If it is, factor it out. AB + ABC → AB(1 + C) → AB 5 See if two of the terms are different by only a NOT function. If it is, factor out all the similar terms. ABC + ABC → AB(C + C) → AB
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Worked Example: Simplify f = AB + ABC + BC. Canonical Forms Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Worked Example: Simplify f = (X + Y )(+Z ). Canonical Forms Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Worked Example: Simplify f = A + BC + AB. Canonical Forms Boolean Simplification
Boolean Theorems Expressing Logic Circuits Analysis of Circuits Worked Example: Simplify F = AB + AC + BC. Canonical Forms Boolean Simplification
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