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Discrete Mathematics and Its Applications

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The English mathematician George Boole (1815-1864) sought to give symbolic form to Aristotle's system of logic. Boole wrote a treatise on the subject in 1854, titled An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities, which codified several rules of relationship between mathematical quantities limited to one of two possible values: true or false, 1 or 0. His mathematical system became known as Boolean algebra. INTRODUCTION

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All arithmetic operations performed with Boolean quantities have but one of two possible outcomes: either 1 or 0. There is no such thing as "2" or "-1" or "1/2" in the Boolean world. It is a world in which all other possibilities are invalid by fiat. As one might guess, this is not the kind of math you want to use when balancing a checkbook or calculating current through a resistor. However, Claude Shannon of MIT fame recognized how Boolean algebra could be applied to on-and-off circuits, where all signals are characterized as either "high" (1) or "low" (0). His 1938 thesis, titled A Symbolic Analysis of Relay and Switching Circuits, put Boole's theoretical work to use in a way Boole never could have imagined, giving us a powerful mathematical tool for designing and analyzing digital circuits. (http://www.databanker.org/Home/BoolAlg/tabid/53/Default.aspx)http://www.databanker.org/Home/BoolAlg/tabid/53/Default.aspx

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The rules of logic are used to design circuits and form the basis for Boolean algebra. The circuits in all electronic devices like computers have inputs that can be identified as either 1 or 0 to produce outputs labeled as 1s and 0s OR switches that are in ‘on’ or in ‘off’ position. Boolean algebra provides the operations and the rules for working with the set {0,1}. The operation of a circuit is defined by a Boolean function that specifies the value of an output for each set of inputs. This function is built using Boolean expressions and operations.

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OPERATIONS IN BOOLEAN ALGEBRA COMPLEMENTATION The complement of an element, denoted with a bar, is defined by: BOOLEAN SUM The Boolean sum, denoted by + or by OR, has the following values: BOOLEAN PRODUCT The Boolean product, denoted by or by AND, has the following values:

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Unless parentheses are used, all complements are computed first followed by Boolean products, then followed by all Boolean sums. The complement, Boolean sum, and the Boolean product correspond to the logical operators,,respectively where 0 is False and 1 is True. Example: Find the value of. Rules of Precedence for Boolean Operators

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BOOLEAN FUNCTION Let Then is the set of all possible n-tuples of 0’s and 1’s. The variable is called a Boolean variable if it assumes values only from B, i.e., if its only possible values are 0 and 1. A function from to is called a Boolean function of degree n.

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Example: The function from the set of ordered pairs of Boolean variables to the set {0,1} is a Boolean function of degree 2 with the values of as shown in the table. 110 101 010 000 Clearly,

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The Boolean expressions in the variables are defined recursively as are Boolean expressions; if and are Boolean expressions, then, and are Boolean expressions. Each Boolean expression represents a Boolean function. The values of this function are obtained by substituting 0 and 1 for the variables in the expression.

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Example: Find the values of the Boolean function represented by 111101 110111 101000 100011 011000 010011 001000 000011 Remark: Two different Boolean expressions that represent the same function are equivalent. Examples: and are equivalent.

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DUALITY and COMPLEMENT The dual of a Boolean expression denoted by, is obtained by interchanging Boolean sums and Boolean products, and interchanging 0’s and 1’s. The dual of the function will be denoted by. Example: Find the dual of and. For, the dual is. The complement of the function is the function That is,

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IDENTITIES 1. Digital computers contain circuits that implement Boolean functions. 2. The simpler that we can make a Boolean function, the smaller the circuit that will result. Simpler circuits are cheaper to build, consume less power, and run faster than complex circuits. 3. With this in mind, we always want to reduce our Boolean functions to their simplest form. 4. There are a number of Boolean identities that help us to do this.

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Most Boolean identities have an AND (product) form as well as an OR (sum) form. We give our identities using both forms.

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Remark: The above identities can be translated to logical equivalences about propositions and to identities about sets.

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We can use Boolean identities to simplify the function as follows:

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Example: Find the complement of the function Applying DeMorgan’s Laws, OR, you can take the complement of the function by taking its dual, then complement each literal (a variable or its complement).

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REPRESENTATION OF BOOLEAN FUNCTIONS PROBLEM: Given the values of a Boolean function, how can a Boolean expression that represents this function be found? Any Boolean function can be represented by a Boolean sum of Boolean products of the variables and their complements. Example: Find Boolean expressions that represents functions and which are given in the following table. 11100 11001 10110 10000 01100 01001 00100 00000

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To represent, we need an expression that has the value 1. A combination of values of the variables for which the function has the value 1 leads to a Boolean product of the variables or their complements. From the table, has the value 1 when and, and the value 0 otherwise. To form the Boolean product that has the value 1, is obtained. What about function ?

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CANONICAL FORMS of a Boolean Function DEFINITION A literal is a Boolean variable or its complement. A minterm is a Boolean product of n literals, with one literal for each variable. A minterm is the standard product. A minterm has the value 1 for one and only combination of values of its variables. Example: Find a minterm that equals 1 if and and equal 0 otherwise. A minterm that has the value 1 is. Given a Boolean function, a Boolean sum of Boolean products (minterms) or simply, sum of minterms, can be formed that has the value 1 when this Boolean function has the value 1, and has the value 0 when the function has the value 0.

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The minterms in this Boolean sum correspond to those combinations of values for which the function has the value 1. The sum of minterms that represents this function is called the sum-of-products expansion or the disjunctive normal form (DNF) of the Boolean function. Example: Find the sum-of-products expansion for the function. The sum-of-products expansion of the function can be done in two ways: 1. Use the Boolean identities to expand the product and simplify.

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2. Construct a table and form the Boolean sum of minterms that have the value 1. Based on the table, the sum-of-terms expansion for the Boolean function is. 111100 110111 101100 100111 011100 01011 11 001000 000010

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It is also possible to get a Boolean expression that represents a Boolean function by taking a Boolean product of Boolean sums (maxterms). The maxterms in this Boolean product correspond to those combinations of values for which the function has the value 0. The resulting expansion is called the product-of-sums expansion or the conjunctive normal form (CNF) of the function. A maxterm is the standard sum. A maxterm has the value 0 for one and only combination of values of its variables. A minterm is the complement of a maxterm, and vice versa. To get the CNF for : 1. Take the complement. 2. Obtain the DNF for. 3. Complement both sides of the DNF obtained in (2). Boolean functions expressed as a sum of minterms (DNF) or product of maxterms (CNF), are said to be in canonical form.

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Minterms & Maxterms for 3 Binary Variables MintermsMaxterms x y zTerm 0 0 0x’y’z’x’y’z’x+y+zx+y+z 0 0 1x’y’zx’y’zx+y+z’ 0 1 0x’yz’x+y’+z 0 1 1x’yzx+y’+z’ 1 0 0xy’z’x’+y+z 1 0 1xy’zx’+y+z’ 1 1 0xyz’x’+y’+z 1 1 1xyzx’+y’+z’

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LOGIC GATES 1. Boolean Algebra is used to model circuitry of electronic devices. 2. Each circuit is designed using the rules of Boolean Algebra. 3. The basic elements of circuits are called gates. 4. Each type of gate implements a Boolean operation. 5. The circuits give output that depends only on the input. 6. Circuits have no memory capabilities. 7. These circuits are called combinational circuits or gating networks and can be constructed using a combination of gates that may share inputs. 8. Each input (entering a gate) and each output (leaving a gate) of a circuit is a member of the set.

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Basic types of Gates xF 01 10 Accepts the value of one Boolean variable as input and produces the complement of this value as output. Inverter or NOT Gate

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AND Gate The inputs to this gate are 2 or more Boolean variables. The output is the Boolean product of their values. xyF 000 010 100 111

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OR Gate The inputs to this gate are the values of 2 or more Boolean variables. The output is the Boolean sum of their values. xyF 000 011 101 111

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NAND Gate xyF 001 011 101 110

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NOR Gate xyF 001 011 101 110

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XOR Gate xyF 000 011 101 110

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XNOR Gate xyF 001 010 100 111

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Discrete Mathematical Structures: Theory and Applications

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Examples 1. Find two ways to draw the same circuit that produces the output 2. Construct circuits that produce the following outputs: a. b. c. 3. Construct a circuit using NAND gates only to produce the following outputs. a. b. c.

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