These forms can simplify the implementation of combinational logic, particularly with PLDs.
Example See the examples in the book ppīoolean expressions can be written in the sum-of-products form (SOP) or in the product-of-sums form (POS). Use the basic laws, rules and theorems of Boolean algebra to manipulate and simplify an expression. Solution Write the expression for each gate: (A + B ) C (A + B ) = C (A + B )+ D X Applying DeMorgan’s theorem and the distribution law: X = C (A B) + D = A B C + Dġ6 Summary Example Simplification using Boolean algebra Example Apply Boolean algebra to derive the expression for X. =ġ5 Summary Example Solution Boolean Analysis of Logic CircuitsĬombinational logic circuits can be analyzed by writing the expression for each gate and combining the expressions according to the rules for Boolean algebra. Solution To apply DeMorgan’s theorem to the expression, you can break the overbar covering both terms and change the sign between the terms. B Applying DeMorgan’s second theorem to gates:ġ4 Summary Example Solution = DeMorgan’s TheoremĪpply DeMorgan’s theorem to remove the overbar covering both terms from the expression X = C + D.
The complement of a sum of variables is equal to the product of the complemented variables. AB = A + B Applying DeMorgan’s first theorem to gates:ġ3 Summary DeMorgan’s Theorem DeMorgan’s 2nd Theorem The complement of a product of variables is equal to the sum of the complemented variables. (A + B)(A + C) = A + BCġ2 Summary DeMorgan’s Theorem DeMorgan’s 1st Theorem ORing with A gives the same area as before. The overlapping area between B and C represents BC. The overlap of red and yellow is shown in orange. The area representing A + C is shown in red. The area representing A + B is shown in yellow. Rule 12, which states that (A + B)(A + C) = A + BC, can be proven by applying earlier rules as follows: (A + B)(A + C) = AA + AC + AB + BC = A + AC + AB + BC = A(1 + C + B) + BC = A BC = A + BC This rule is a little more complicated, but it can also be shown with a Venn diagram, as given on the following slide…ġ1 Summary Three areas represent the variables A, B, and C. A + AB = A + B Example Solution This time, A is represented by the blue area and B again by the red circle. Other rules can be illustrated with the diagrams as well.ĩ Summary Example Solution Rules of Boolean Algebra = The diagram visually shows that A + AB = A. The overlap region between A and B represents AB. Add an overlapping area to represent the variable B. The rule A + AB = A can be illustrated easily with a diagram. Rules of Boolean algebra can be illustrated with Venn diagrams. That is AB + AC = A(B+ C) The distributive law can be illustrated with equivalent circuits: A(B+ C) AB + ACħ Summary Rules of Boolean Algebra 1. A common variable can be factored from an expression just as in ordinary algebra. The distributive law is the factoring law. A + (B +C) = (A + B) + C For multiplication, the associative law states When ANDing more than two variables, the result is the same regardless of the grouping of the variables. For addition, the associative law states When ORing more than two variables, the result is the same regardless of the grouping of the variables. The associative laws are also applied to addition and multiplication. A + B = B + A For multiplication, the commutative law states In terms of the result, the order in which variables are ANDed makes no difference. For addition, the commutative law states In terms of the result, the order in which variables are ORed makes no difference. The commutative laws are applied to addition and multiplication. What are the values of the A, B and C if the product term of A.B.C = 1? Example Solution Each literal must = 1 therefore A = 1, B = 0 and C = 0. The product term will be 1 only if all of the literals are 1. The product of literals forms a product term. In Boolean algebra, multiplication is equivalent to the AND operation. Determine the values of A, B, and C that make the sum term of the expression A + B + C = 0? Example Solution Each literal must = 0 therefore A = 1, B = 0 and C = 1.ģ Summary Example Solution Boolean Multiplication The sum term is zero only if each literal is 0. The sum term is 1 if one or more if the literals are 1. Addition is equivalent to the OR operation. A literal is a variable or its complement. The complement represents the inverse of a variable and is indicated with an overbar. A single variable can only have a value of 1 or 0. In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. 1 Digital Fundamentals Floyd Chapter 4 Tenth EditionĢ Summary Example Solution Boolean Addition
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