# Lecture (04) Boolean Algebra and Logic Gates

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1 Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Boolean algebra properties basic assumptions and properties: Closure law A set S is closed with respect to a binary operator, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S. ٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

2 Commutative law. A binary operator * on a set S is said to be commutative whenever ٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Associative law. A binary operator * on a set S is said to be associative whenever ٤ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

3 ٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Identity element. A set S is said to have an identity element with respect to a binary operation * on S if there exists an element e S with the property that Example: The element is an identity element with respect to the binary operator + on the set of integers I = {, 3, 2,,,, 2, 3, }, since x + = + x = x for any x I Example: The element is an identity element with respect to the binary operator. on the set of integers I = {, 3, 2,,,, 2, 3, }, since x. =. x = x for any x I

4 ٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Inverse. A set S having the identity element e with respect to a binary operator * is said to have an inverse whenever, for every x S, there exists an element y S such that Example: In the set of integers, I, and the operator +, with e =, the inverse of an element a is ( a), since a + ( a) =. ٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

5 ٩ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Distributive law If * and. are two binary operators on a set S, * is said to be distributive over. Whenever ١٠ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

6 ١١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Summery The binary operator + defines addition (OR). The additive identity is. The additive inverse defines subtraction. The binary operator. defines multiplication (AND). The multiplicative identity is. For a, the multiplicative inverse of a = /a defines division (i.e., a. /a = ). The only distributive law applicable is that of. over +: a. (b + c) = (a. b) + (a.. c) ١٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

7 Ordinary and Boolean Algebra George Boole developed an algebraic system now called Boolean algebra. Claude E. Shannon introduced a two valued Boolean algebra called switching algebra that represented the properties of bistable electrical switching circuits. ١٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design. (a) The structure is closed with respect to the operator +. (b) The structure is closed with respect to the operator.. 2. (a) The element is an identity element with respect to +; that is, x + = + x = x. (b) The element is an identity element with respect to. ; that is, x. =. x = x. 3. (a) The structure is commutative with respect to +; that is, x + y = y + x. (b) The structure is commutative with respect to. ; that is, x. y = y. x. ١٤ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

8 4. (a) The operator. is distributive over +; that is, x. (y + z) = (x. y) + (x. z). (b) The operator + is distributive over. ; that is, x + (y. z) = (x + y). (x + z). 5. For every element x B, there exists an element x B (called the complement of x) such that (a) x + x = and (b) x. x =, 6. There exist at least two elements x, y B such that x y. ١٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Comparing Boolean algebra with arithmetic and ordinary algebra: The distributive law of + over. (i.e., x + (y. z) = (x + y). (x + z) ) is valid for Boolean algebra, but not for ordinary algebra. Boolean algebra does not have additive or multiplicative inverses; therefore, there are no subtraction or division operations. defines an operator called the complement that is not available in ordinary algebra.

9 Ordinary algebra deals with the real numbers, which constitute an infinite set of elements. Boolean algebra deals with the as yet undefined set of elements, B, but in the two valued Boolean algebra defined next (and of interest in our subsequent use of that algebra), B is defined as a set with only two elements, and. ١٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Two Valued Boolean Algebra defined on a set of two elements, B = {, }, with rules for the two binary operators + and. Truth table rules are exactly the same as the AND, OR, and NOT operations ١٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

10 the structure is closed with respect to the two operators, each operation is either or and, B. From the tables, we see that (a) + = + = + = ; (b). =. =. =. This establishes the two identity elements, for + and for., The commutative laws are obvious from the symmetry of the binary operator table (A+B = B+A) & (A.B = B.A) ١٩ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design The distributive law x. (y + z) = (x. y) + (x. z) can be shown to hold from the operator tables The distributive law of + over. can be shown to hold by means of a truth table ٢٠ Truth table Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

11 the complement table (a) x + x =, since + = + = and + = + =. (b) x. x =, since. =. = and. =. =. ٢١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Postulates and Basic theorem of Boolean algebra Postulates and Theorems of Boolean Algebra ٢٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

12 Duality principle postulates were listed in pairs and designated by part (a) and part (b). One part may be obtained from the other if the binary operators and the identity elements are interchanged we simply interchange OR and AND operators and replace s by s and s by s. ٢٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Theorem 5.a Truth table ٢٤ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

13 Truth table ٢٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Operator Precedence () parentheses, (2) NOT, (3) AND, and (4) OR. expressions inside parentheses must be evaluated before all other operations. The next operation that holds precedence is the complement, and then follows the AND and, finally, the OR. example: demorgan s ٢٦ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

14 Boolean functions Boolean function described by an algebraic expression consists of binary variables A Boolean function can be represented in a truth table. The number of rows in the truth table is 2 n, where n is the number of variables in the function. The interconnection of gates will dictate the logic expression. ٢٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Logic diagram ٢٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

15 Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٢٩ F x y z Y' z y x it is sometimes possible to obtain a simpler expression for the same function and thus reduce the number of gates in the circuit and the number of inputs to the gate. Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٠

16 Logic diagram ٣١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Logic diagram ٣٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

17 Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٣ z y x f2 X z z X Xy Y X Truth table Algebraic Manipulation When a Boolean expression is implemented with logic gates, each term requires a gate and each variable within the term designates an input to the gate. We define a literal to be a single variable within a term, in complemented or un complemented form. By reducing the number of terms, the number of literals, or both in a Boolean expression, it is often possible to obtain a simpler circuit Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٤

18 Example ٣٥ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٦ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

19 Complement of a Function The complement of a function may be derived algebraically through DeMorgan s theorems DeMorgan s theorems can be extended to three or more variables. ٣٧ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٣٨ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

20 Example 2 ٣٩ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٤٠ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

21 Example 3 ٤١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design ٤٢ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

22 Thanks,.. See you next week (ISA), ٤٣ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design

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