BOOLEAN ALGEBRA. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits.

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1 COURSE / CODE DIGITAL SYSTEMS FUNDAMENTAL (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) BOOLEAN ALGEBRA Boolean Logic Boolean logic is a complete system for logical operations. It is used in countless systems. It was named after George Boole, who first defined an algebraic system of logic in the mid 19th century. Boolean logic has many applications in electronics, computer hardware and software, and is the base of all modern digital electronics. In 1938, Claude Shannon showed how electric circuits with relays were a model for Boolean logic. This fact soon proved enormously consequential with the emergence of the electronic computer. Boolean logic is also used for circuit design in electrical engineering; here 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression. Basic logic gates such as AND, OR, and NOT gates may be used alone, or in conjunction with NAND, NOR, and XOR gates, to control digital electronics and circuitry. Whether these gates are wired in series or parallel controls the precedence of the operations. Truth Table A truth table is a mathematical table used in logic specifically in connection with Boolean algebra, Boolean functions, and propositional calculus to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables. In particular, truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is, logically valid. Practically, a truth table is composed of one column for each input variable (for example, A and B), and one final column for all of the possible results of the logical operation that the table is meant to represent (for example, A XOR B). Each row of the truth table therefore contains one possible configuration of the input variables (for instance, A=true B=false), and the result of the operation for those values. Logic circuit: 1. From logic circuit to Boolean expression. Derive the Boolean expression for the following circuits. 2. From Boolean expression to logic circuit. Draw the logic circuit for the following expression. a) F W X YZ WY Z b) F ABCD ABCD ABCD ABCD Mohd Uzir Kamaluddin / Aug 2016 page 1

2 Boolean Algebra Boolean algebra developed in 1854 by George Boole in his book An Investigation of the Laws of Thought, is a variant of ordinary algebra as taught in high school. Boolean algebra differs from ordinary algebra in three ways: in the values that variables may assume, which are of a logical instead of a numeric character, prototypically 0 and 1; in the operations applicable to those values; and in the properties of those operations, that is, the laws they obey. Prove the following: 1. A+AB = A 2. A(A+B) = A 3. A+ A B = A+B 4. A( A +B) = AB 5. ( A +B)(A+B) = B Exercises (notation used a = a ) 1. What function is implemented by the circuit shown a) x'y'+z b) (x'+y')z c) x'y'z d) x'+y'+z e) Not Available 2. What function is implemented by the circuit shown a) x+y+z b) x+y+z' c) x'y'z d) x'+y'+z' e) Not Available 3. What function is implemented by the circuit shown a) xz'+y b) xz+y c) x'z+y' d) x'y'+y'z' e) x'y'+y'z Mohd Uzir Kamaluddin / Aug 2016 page 2

3 4. Which gate is the following circuit equivalent to? a) AND b) OR c) NAND d) NOR e) None of the above 5. Which of the following functions equals the function: f=x+yz'? a) x(y'+z) d) (y+x')(x'+z') b) x(y'+z) e) Not Available c) (y+x)(z'+x) De Morgan s Theorem This theorem basically states that: 1. the complement of the product of a given set of variables is equal to the sum of the complements of the individual variables; and 2. the complement of the sum of a given set of variables is equal to the product of the complements of the individual variables. De Morgan's Theorem applies to any arbitrary number of variables. The important implication of De Morgan's Theorem is that any logic gate or circuit can be replaced by an equivalent circuit composed of other gates, as long as the NOT function can be provided by at least one of the substitute gates. This is useful in digital electronics design wherein there is a need to minimize the variety and number of logic gate IC's used. Two forms of De Morgan's Theorem implemented with basic gates. Simplify the following logic circuits. Mohd Uzir Kamaluddin / Aug 2016 page 3

4 Convert a logic circuit into NAND gates only The NAND gate is called a universal gate because combinations of it can be used to accomplish all the basic functions. The following logic circuits are the same. Convert the following logic circuit into its NAND equivalent. Exercises (notation used a = a ) 1. Any possible binary logic function can be implemented using only. a) AND d) AA (anyone is sufficient) b) OR e) NAND c) NOT 2. The function in the following circuit is: a) abcd b) ab+cd c) (a+b)(c+d) d) a+b+c+d e) (a'+b')(c'+d') Mohd Uzir Kamaluddin / Aug 2016 page 4

5 3. Given F=A'B+(C'+E)(D+F'), use de Morgan's theorem to find F'. a) ACE'+BCE'+D'F d) ACE'+AD'F+B'CE'+B'D'F b) (A+B')(CE'D'F) e) Not available c) A+B+CE'D'F 4. The function in the following circuit is: a) x'+y'+z' b) x+y+z c) x'z'+y'z' 5. Simplify the following: {[(AB)'C]'D}' a) (A'+B')C+D' b) (A+B')C'+D' c) A'+(B'+C')D d) xy+z e) z d) A'+B'+C'+D' e) A+B+C+D Sum-of-Products Expression (SOP) A sum-of-products (SOP) expression is a Boolean expression in a specific format. The term sum-ofproducts comes from the expression's form: a sum (OR) of one or more products (AND). As a digital circuit, an SOP expression takes the output of one or more AND gates and OR's them together to create the final output. The inputs to the AND gates are either inverted or non-inverted input signals. This limits the number of gates that any input signal passes through before reaching the output to an inverter, an AND gate, and an OR gate. Since each gate causes a delay in the transition from input to output, and since the SOP format forces all signals to go through exactly two gates (not counting the inverters), an SOP expression gives us predictable performance regardless of which input in a combinational logic circuit changes. This is an example of an SOP expression: ABCD ACD AB CD There are no parentheses in an SOP expression since they would necessitate additional levels of logic. This also means that an SOP expression cannot have more than one variable combined in a term with an inversion bar. Converting an SOP expression to a Truth Table To convert an SOP expression to a truth table, examine each product to determine when it is equal to 1. Where there that product is a 1, a 1 will also be outputted from the circuit. Mohd Uzir Kamaluddin / Aug 2016 page 5

6 Converting a Truth Table to an SOP expression Any truth table can be converted to an SOP expression. The conversion process goes like this: identify the rows with ones as the output, and then come up with the unique product to put a one in that row. Note that this will give us an SOP expression where all of the products use all of the variables for inputs. This usually gives us an expression that can be simplified. OR ing all these products together will give the SOP expression: X ABC ABC ABC ABC The short hand form, X can be written as: X m (1,3,4,6 ) Product-of-Sums Expression (POS) The product-of-sums (POS) format of a Boolean expression is much like the SOP format with its two levels of logic (not counting inverters). The difference is that the outputs of multiple OR gates are combined with a single AND gate which outputs the final result. This expression adheres to the format of a POS expression. ( A B C)( A B C)( A B C) Converting a POS expression to Truth Table Converting a POS expression to a truth table follows a similar process as the one used to convert an SOP expression to a truth table. The difference is this: where the SOP conversion focuses on rows with a 1 output, the POS conversion focuses on rows with a 0 output. Mohd Uzir Kamaluddin / Aug 2016 page 6

7 Converting a Truth Table to a POS expression Just as with SOP expressions, any truth table can be converted to a POS expression. The conversion process goes like this: identify the rows with zeros as the output, and then come up with the unique sum to put a zero in that row. The final group of sums can then be AND'ed together producing the POS expression. AND ing all these sums together will give the POS expression: X ( A B C)( A B C)( A B C)( A B C) The short hand form, X can be written as: X M (0,2,5,7 ) Minimization of Logic Functions Many times in the application of Boolean Algebra, a particular expression need to be reduced to its simplest form or change its form to a more convenient one to implement the expression efficiently. 1) By using Boolean algebra Minimize the following functions to its simplest form: a) F = AB CD BCD AD ACD ABCD b) F = AB + A(B + C) + B(B + C) c) F = ( A B( C BD) AB) C d) F = ABC ABC ABC ABC ABC Boolean Algebra has the disadvantage of being unsystematic in solving expressions and also it is unknown whether the expression has reached its simplest form. 2) By using K-map method The Karnaugh map, also known as a Veitch diagram (K-map or KV-map for short), is a tool to facilitate management of Boolean algebraic expressions. A Karnaugh map is unique in that only one variable changes value between squares, in other words, the rows and columns are ordered according to the principles of Gray code. Mohd Uzir Kamaluddin / Aug 2016 page 7

8 In a Karnaugh map with n variables, a Boolean term mentioning k of them will have a corresponding rectangle of area 2n k. Common sized maps are of 2 variables which is a 2x2 map; 3 variables which is a 2x4 map; and 4 variables which is a 4x4 map. Karnaugh Maps - Rules of Simplification The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing 1s. Groups may not include any cell containing a 0. Groups may be horizontal or vertical, but not diagonal. Groups must contain 1, 2, 4, 8, or in general 2n cells. That is if n = 1, a group will contain two 1's since 21 = 2. If n = 2, a group will contain four 1's since 22 = 4. Each group should be as large as possible. Mohd Uzir Kamaluddin / Aug 2016 page 8

9 Each cell containing a 1 must be in at least one group. Groups may overlap. Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell. There should be as few groups as possible, as long as this does not contradict any of the previous rules. Example 1: Design a combinational logic circuit with three inputs and one output. The output is a 1 when the binary value of the input is less than three. Otherwise the output is 0. Mohd Uzir Kamaluddin / Aug 2016 page 9

10 Example 2: A logic circuit that has three inputs A, B, C and one output F has the following truth table. Input A B C Output F Write down the Boolean function of the truth table as sum-of-products (SOP) and product-of-sum (POS) forms, then simplify it using K-map for both the SOP and POS expressions and draw the logic circuit. The K-map is: AB C To get the SOP expression, read the 1 s in groups. F AB AB BC or F AB AB AC To get the POS expression, read the 0 s in groups (care!) F ( A B)( A B C) Exercise 2: A logic circuit that has three inputs A, B, C and one output F. The output F is high only when the majority of its input is high. Implement the logic circuit. Exercise 3: Simplify the following Boolean functions, using a K-map. a) X m (0,1,2,4,6 ) b) X M (0,2,4,5,6 ) c) AB AC BC ABC Exercise 4. Using K-map, minimize the following Boolean expressions: a) AB ( A B) C AB b) Z = f(a,b,c) = + B + AB + AC c) Z = f(a,b,c) = B + B + BC + A Mohd Uzir Kamaluddin / Aug 2016 page 10

11 Exercise 5. Reduce the following logic circuit into its simplest form. A B C D X Exercise 6. Simplify the following logic circuit into its simplest form by the use of Boolean Algebra and then implement the resulting circuit. Exercise 7. a) Examine the circuit shown below and determine the Boolean equation representing the circuit. b) Simplify the Boolean equation using (i) Boolean reduction and (ii) Karnaugh mapping. c) Draw a truth-table for the circuit. d) Perform a logic simulation of the circuit using Logisim and compare the results with the truth table obtained earlier. e) Construct the circuit as shown and verify that it functions according to the truth-table. f) Using the results from (b) design the simplest circuit which performs the same function as the original circuit. (The minimal solution consists of 2 gates.) g) Perform a logic simulation of the simplified circuit in order to verify the design. h) Finally, construct the simplified solution and confirm that it is correct. Mohd Uzir Kamaluddin / Aug 2016 page 11

12 Four variables K-map For the following K-maps, find their minimal SOP expressions. Example 1. Minimize the following expressions using four variable K-Map. a) F(x,y,w,z) = Σm(0,1,2,3,4,6,11,14) b) F(x,y,w,z) = Σm (0,2,4,6,12,14) c) F(x,y,w,z) = Σm (0,2,5,7,8,11,13,15) Example 2. Draw the truth table with inputs D, C, B, A and output F. If the input DCBA is a prime number, then F is high, otherwise low. The prime numbers in the range 0 to 15 are 3, 5, 7, 11, 13. The truth table is given below. Using a K-Map, find the simplest expression for F. Mohd Uzir Kamaluddin / Aug 2016 page 12

13 Example 3. Given the following truth table, find the simplest expression for the output. Example 4. Draw the truth table for 4 bit Binary (B3,B2,B1,B0) to Gray code (G3,G2,G1,G0). Obtain the simplest expression for the outputs using K-maps and draw the resulting circuit diagram. A good website that explain and solve K-maps online:- Mohd Uzir Kamaluddin / Aug 2016 page 13

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