De Morgans Theorem Theorems of Boolean Algebra 1

De Morgan’s Theorem,

Theorems of Boolean Algebra 1) A + 0 = A 2) A + 1 = 1 3) A • 0 = 0 4) A • 1 = A 5) A + A = A 6) A + A = 1 7) A • A = A 8) A • A = 0

Theorems of Boolean Algebra 9) A = A 10) A + AB = A 11) A + AB = A + B 12) (A + B)(A + C) = A + BC 13) Commutative : A + B = B + A AB = BA 14) Associative : A+(B+C) =(A+B) + C A(BC) = (AB)C 15) Distributive : A(B+C) = AB +AC (A+B)(C+D)=AC + AD + BC + BD

De Morgan’s Theorems • Two most important theorems of Boolean Algebra were contributed by De Morgan. • Extremely useful in simplifying expression in which product or sum of variables is inverted. • The TWO theorems are : 16) (X+Y) = X. Y 17) (X. Y) = X + Y

Implications of De Morgan’s Theorem (a) (b) Input Output X Y X+Y XY 0 0 1 1 0 0 0 2 1 1 0 (c) 0 (a) Equivalent circuit implied by theorem (16) (b) Alternative symbol for the NOR function (c) Truth table that illustrates De. Morgan’s Theorem

Implications of De Morgan’s Theorem (a) (b) Input Output X Y XY X+Y 0 0 1 1 1 1 0 1 1 2 1 1 0 (c) 0 (a) Equivalent circuit implied by theorem (17) (b) Alternative symbol for the NAND function (c) Truth table that illustrates De. Morgan’s Theorem

De Morgan’s Theorem Conversion Step 1: Change all ORs to ANDs and all ANDs to Ors Step 2: Complement each individual variable (short overbar) Step 3: Complement the entire function (long overbars) Step 4: Eliminate all groups of double overbars Example : = = A. B A+B A+B A. B. C = A+B+C

De Morgan’s Theorem Conversion = = ABC + ABC (A+B+C) (A + B +C)D = (A. B. C)+D

Examples: Analyze the circuit below Y 1. Y=? ? ? 2. Simplify the Boolean expression found in 1

• Follow the steps list below (constructing truth table) – List all the input variable combinations of 1 and 0 in binary sequentially – Place the output logic for each combination of input – Base on the result found write out the boolean expression.

Exercises: • • Simplify the following Boolean expressions 1. (AB(C + BD) + AB)C 2. ABC + ABC Write the Boolean expression of the following circuit.

Standard Forms of Boolean Expressions Sum of Products (SOP) Products of Sum (POS) Notes: § SOP and POS expression cannot have more than one variable combined in a term with an inversion bar § There’s no parentheses in the expression

Standard Forms of Boolean Expressions Converting SOP to Truth Table § Examine each of the products to determine where the product is equal to a 1. § Set the remaining row outputs to 0.

Standard Forms of Boolean Expressions Converting POS to Truth Table § Opposite process from the SOP expressions. § Each sum term results in a 0. § Set the remaining row outputs to 1.

Standard Forms of Boolean Expressions The standard SOP Expression § All variables appear in each product term. § Each of the product term in the expression is called as minterm. § Example: § In compact form, f(A, B, C) may be written as

Standard Forms of Boolean Expressions The standard POS Expression § All variables appear in each product term. § Each of the product term in the expression is called as maxterm. § Example: § In compact form, f(A, B, C) may be written as

Standard Forms of Boolean Expressions Example: Convert the following SOP expression to an equivalent POS expression: Example: Develop a truth table for the expression: