Slide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary

Save this PDF as:

Size: px
Start display at page:

Download "Slide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary" Transcription

1 Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016

2 SN s ENEL 353 Fall 2016 Slide Set 3 slide 2/49 Contents Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

3 SN s ENEL 353 Fall 2016 Slide Set 3 slide 3/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

4 SN s ENEL 353 Fall 2016 Slide Set 3 slide 4/49 Ordinary Algebra We re very accustomed to seeing things like (a + b)(c + d) = ac + ad + bc + bd and If x 2 + x = 0 and x 0, then x + 1 = 0. Symbols such as a, b, c, d, and x are variables that could have any value taken from a set such as the set of all real numbers, or the set of all complex numbers.

5 SN s ENEL 353 Fall 2016 Slide Set 3 slide 5/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

6 SN s ENEL 353 Fall 2016 Slide Set 3 slide 6/49 Boolean Algebra In Boolean algebra, variables have values taken from this set: {0, 1}. Ordinary algebra has many operators: addition, subtraction, multiplication, division, and others. Boolean algebra has three operators: NOT, AND, and OR. Boolean algebra is very useful for description, analysis and design of combinational logic systems. Note: Older textbooks for ENEL 353 use the term switching algebra instead of Boolean algebra. Watch out for that if you are looking at previous years course materials.

7 SN s ENEL 353 Fall 2016 Slide Set 3 slide 7/49 Terminology for Boolean algebra There are many words that have special meanings in discussion of Boolean algebra. Here are some of them: complement, literal, product, sum, minterm, maxterm. There are several others. Make sure you learn exact meanings for all of them! Having only a rough idea what they mean is not good enough that will lead to confusion and errors.

8 SN s ENEL 353 Fall 2016 Slide Set 3 slide 8/49 Variables, complements and literals As stated two slides back, variables in Boolean algebra are things that may have one of two values: 0 (FALSE), or 1 (TRUE). The complement of a variable is the NOT of that variable. So the complement of A is A, the complement of B is B, and so on. A literal is either a variable or the complement of a variable. So, if A and B are Boolean variables, which of the following expressions are literals? A A B B AB A + B A + A

9 SN s ENEL 353 Fall 2016 Slide Set 3 slide 9/49 True form and complementary form These are terms that distinguish the two kinds of literals. The true form of a variable is just the plain version of that variable. For example, the true forms of A, B, etc., are simply A, B, etc. The complementary form of a variable is the NOT of that variable. For example, the complementary forms of A, B, and so on, are A, B, and so on.

10 SN s ENEL 353 Fall 2016 Slide Set 3 slide 10/49 Products In Boolean algebra, a product is defined to be either a literal; or the AND of two or more literals. (Our textbook defines a product as the AND of one or more literals, which is correct, but makes you think harder than necessary about what the AND of one literal is.) Suppose A, B, and C are Boolean variables. Which of the following expressions are products? A AB ABC C A + B AB + BC

11 SN s ENEL 353 Fall 2016 Slide Set 3 slide 11/49 Minterms Suppose a function has N input variables. A minterm is defined to be a product that uses all N of the input variables each variable appearing once in either true or complemented form. For example, if the input variables are A and B, there are 4 minterms: Ā B, ĀB, A B, and AB. If the input variables are A, B and C, how many minterms are there, and what are the minterms?

12 SN s ENEL 353 Fall 2016 Slide Set 3 slide 12/49 Sums and maxterms A sum is either a literal; or two or more literals OR-ed together. Examples of sums: A, B, A + B, A + B + C. Examples of expressions that are not sums: AB, A + C, A + B C. A maxterm is defined to be a sum that uses all N of the input variables each variable appearing once in either true or complementary form. (This definition differs from the minterm definition by only one word!) For input variables A and B, what are all the maxterms?

13 SN s ENEL 353 Fall 2016 Slide Set 3 slide 13/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

14 SN s ENEL 353 Fall 2016 Slide Set 3 slide 14/49 Order of operations in Boolean algebra Order of operations is related to the concept of operator precedence. In Boolean algebra: NOT has highest precedence. AND is next. OR has lowest precedence. Let s use parentheses to show order of operations in the following expressions. AB + BC ĀB + B C

15 SN s ENEL 353 Fall 2016 Slide Set 3 slide 15/49 Clarification about the notation for NOT NOT is a unary operator that means NOT applies to a single operand. So when you see an overline over a complex expression, that expression has to be evaluated before NOT can be applied. So, for example, AB really means (AB) ; A + B really means (A + B) ; ĀB + A B really means ( ĀB + A B ).

16 SN s ENEL 353 Fall 2016 Slide Set 3 slide 16/49 Associativity of AND and OR operators We ve just seen expressions such as Ā B C ; A + B + C. These might raise questions, such as Does the first example mean ( Ā B ) C or Ā ( B C), or does it matter? Does the second example mean (A + B) + C or A + ( B + C ), or does it matter? In fact, AND and OR are both associative, which means that (XY )Z = X (YZ) it doesn t matter which AND is done first. So we usually just write XYZ. (X + Y ) + Z = X + (Y + Z), so we usually just write X + Y + Z.

17 SN s ENEL 353 Fall 2016 Slide Set 3 slide 17/49 Let s prove that (XY )Z = X (YZ). (X + Y ) + Z = X + (Y + Z) can be proved in a similar manner. Recall that the N-input AND function was defined as { 1 if all of A1, A AND (A 1, A 2,..., A N ) = 2,..., A N are 1 0 otherwise Because the AND operator is associative, we can write AND (A 1, A 2,..., A N ) = A 1 A 2 A N. Similarly for N-input OR, we can write OR (A 1, A 2,..., A N ) = A 1 + A A N.

18 SN s ENEL 353 Fall 2016 Slide Set 3 slide 18/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

19 SN s ENEL 353 Fall 2016 Slide Set 3 slide 19/49 Sum-of-products expressions A sum-of-products often abbreviated as SOP is defined to be either a single product, or two or more products, ORed together. With variables A, B and C, which of the following are SOP expressions? 1. ĀC 2. A + B + C 3. A + ĀBC 4. Ā BC + ĀB C + A B C + ABC 5. A( B C + BC) 6. AB + BC

20 SN s ENEL 353 Fall 2016 Slide Set 3 slide 20/49 Sum-of-products canonical form Sum-of-products canonical form is defined to be an SOP expression for a Boolean function in which all the products are minterms. With variables A, B and C, which of the following expressions are in SOP canonical form? 1. ĀC 2. A + B + C 3. A + ĀBC 4. Ā BC + ĀB C + A B C + ABC 5. A( B C + BC) 6. AB + BC

21 SN s ENEL 353 Fall 2016 Slide Set 3 slide 21/49 Using SOP canonical form to get a Boolean equation from a truth table Suppose you have a truth table for some function F, which is a function of some number of Boolean variables. Then there is a straightforward procedure for finding the SOP canonical form equation for F. Step 1: For each row in the table that has F = 1, find which minterm is true for that row. Step 2: OR together all the minterms found in Step 1. Let s use this procedure to find SOP canonical form equations for F = A B (the XNOR function), and the sum function of a 1-bit full adder.

22 SN s ENEL 353 Fall 2016 Slide Set 3 slide 22/49 Minterm numbering Consider a truth table involving N variables. We can number the rows of the table starting at 0 and ending with 2 N 1. The number of a row will be the value of the unsigned binary number made up of the 0 s and 1 s in that row on the input side of the table. m k, read as minterm k, is the minterm that is true for the input values in row k of the table. Let s work out the minterm numbering for the cases of (a) two input variables and (b) three input variables.

23 SN s ENEL 353 Fall 2016 Slide Set 3 slide 23/49 Minterm numbering and shorthand for SOP canonical form There are a couple of compact ways to use minterm numbers to write SOP canonical forms. These avoid writing out all the literals involved, and are expecially handy for functions of 3 or more variables with lots of minterms. For example, if SOP canonical form for F (A,B,C) is the OR of minterms m 0, m 1, m 3, m 4, and m 6, we can write F = Σ(m 0, m 1, m 3, m 4, m 6 ) or just F = Σ(0,1,3,4,6).

24 SN s ENEL 353 Fall 2016 Slide Set 3 slide 24/49 Minterm numbering and shorthand for SOP canonical form: Examples Let s write the SOP canonical form for F (A,B) = A B using minterm numbers. Let s write the SOP canonical form for the sum function of a one-bit full adder using minterm numbers. For the function F = Σ(m 0, m 1, m 3, m 4, m 6 ), let s determine the truth table, assuming that the input variables are A, B, and C.

25 SN s ENEL 353 Fall 2016 Slide Set 3 slide 25/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

26 SN s ENEL 353 Fall 2016 Slide Set 3 slide 26/49 Products of sums and product-of-sums canonical form A product-of-sums often abbreviated as POS is defined to be either a single sum, or two or more sums, ANDed together. With variables A, B and C, which of the following are POS expressions? A + B (A + B)(A + C) A(B + C) A BC An expression in product-of-sums canonical form is defined to be a POS expression in which all the sums are maxterms.

27 SN s ENEL 353 Fall 2016 Slide Set 3 slide 27/49 From truth table to SOP or POS canonical form The table below summarizes a method to find SOP canonical form from a truth table this is review of material on slide 21; a method to find POS canonical form from a truth table. Goal Step 1 Step 2 SOP canonical For truth table rows OR together form with F = 1, find true minterms found in POS canonical form minterms. For truth table rows with F = 0, find false maxterms. Step 1. AND together maxterms found in Step 1.

28 SN s ENEL 353 Fall 2016 Slide Set 3 slide 28/49 From truth table to POS canonical form Goal Step 1 Step 2 POS canonical form For truth table rows with F = 0, find false maxterms. AND together maxterms found in Step 1. Let s apply this method to find POS canonical form for F = A B. Let s apply this method to find POS canonical form for the sum function of a 1-bit full adder.

29 SN s ENEL 353 Fall 2016 Slide Set 3 slide 29/49 Remark about methods for finding SOP and POS canonical forms There is some simple intuition behind the method for SOP canonical form: Really, you re just making a list of all of the different combinations of input bits that make the output function true. Your instructor s opinion about the method for POS canonical form: Maybe there s an intuitive way to understand it, but if so, it s not very simple!

30 SN s ENEL 353 Fall 2016 Slide Set 3 slide 30/49 Maxterm numbering and shorthand for POS canonical forms The number of a maxterm is the row number of the truth table row for which that maxterm is false. Let s find maxterm numbers for a function of 3 variables, then use those to write POS canonical form for the sum-output-of-1-bit-full-adder example. M k is read as maxterm k. Attention: Use lowercase m for minterm, and uppercase M for maxterm. Π notation (Π is the uppercase of π) is sometimes used to represent a product of maxterms. Let s do an example.

31 SN s ENEL 353 Fall 2016 Slide Set 3 slide 31/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

32 SN s ENEL 353 Fall 2016 Slide Set 3 slide 32/49 Axioms of Boolean algebra The axioms of a mathematical system are a minimal set of basic definitions we assume to be true. Everything true about the system can be derived from its axioms. For Boolean algebra, the axioms are: (A1) If B 1 then B = 0. (A1 ) If B 0 then B = 1. (A2) 0 = 1 (A2 ) 1 = 0 (A3) 0 0 = 0 (A3 ) = 1 (A4) 1 1 = 1 (A4 ) = 0 (A5) 0 1 = 1 0 = 0 (A5 ) = = 1 Question: Does the above table convey any new information, information that we have not already seen in ENEL 353?

33 SN s ENEL 353 Fall 2016 Slide Set 3 slide 33/49 Theorems of Boolean algebra Theorems of Boolean algebra are relations between expressions in Boolean algebra; the theorems can be derived from the axioms. Theorems can be used to transform an expression into a different-looking but equivalent expression. Often (but not always) the goal of transformation is to simplify an expression for a Boolean function in some sense to go from a complicated description of the function to a simpler one. (Note: Theorems of Boolean algebra have practical value, but they aren t as deep and interesting, as, say, the important theorems in differential and integral calculus!)

34 SN s ENEL 353 Fall 2016 Slide Set 3 slide 34/49 Theorems of One Variable These are all fairly obvious consequences of how the AND, OR, and NOT operations are defined... Theorem Dual Theorem Name (T1) B 1 = B (T1 ) B + 0 = B Identity (T2) B 0 = 0 (T2 ) B + 1 = 1 Null Element (T3) B B = B (T3 ) B + B = B Idempotency (T4) B = B Involution (T5) B B = 0 (T5 ) B + B = 1 Complements Every one of these nine theorems is sometimes handy in algebraic manipulation of Boolean expressions.

35 SN s ENEL 353 Fall 2016 Slide Set 3 slide 35/49 Dual Axioms and Theorems Every axiom or theorem in which 0, 1, AND or OR is present has a dual an axiom or theorem obtained by making all for these substitutions: 1 for 0; 0 for 1; OR for AND; AND for OR. You can see this principle at work in the table of theorems of one variable... Theorem Dual Theorem Name (T1) B 1 = B (T1 ) B + 0 = B Identity (T2) B 0 = 0 (T2 ) B + 1 = 1 Null Element (T3) B B = B (T3 ) B + B = B Idempotency (T4) B = B Involution (T5) B B = 0 (T5 ) B + B = 1 Complements

36 SN s ENEL 353 Fall 2016 Slide Set 3 slide 36/49 Some theorems of several variables The textbook assigns numbers to several-variable theorems from T6/T6 to T12/T12. Here is a partial table: Theorem and Dual Theorem Name T6 B C = C B Commutativity T6 B + C = C + B T7 (B C) D = B (C D) Associativity T7 (B + C) + D = B + (C + D) T8 (B C) + (B D) = B (C + D) Distributivity T8 (B + C) (B + D) = B + (C D) T6 and T6 are obvious from the truth tables for AND and OR. T7 and T7 are the associativity properties of AND and OR mentioned previously. (We actually proved T7 by perfect induction.)

37 SN s ENEL 353 Fall 2016 Slide Set 3 slide 37/49 Distributivity properties Let s review the table from the previous slide... Theorem and Dual Theorem Name T6 B C = C B Commutativity T6 B + C = C + B T7 (B C) D = B (C D) Associativity T7 (B + C) + D = B + (C + D) T8 (B C) + (B D) = B (C + D) Distributivity T8 (B + C) (B + D) = B + (C D) Let s look at Theorems T8 and T8, and compare them to similar-looking equations in ordinary algebra.

38 SN s ENEL 353 Fall 2016 Slide Set 3 slide 38/49 More several-variable theorems Theorem and Dual Theorem Name T9 B (B + C) = B Covering T9 B + (B C) = B T10 (B C) + (B C) = B Combining T10 (B + C) (B + C) = B Let s prove T10 two ways: by perfect induction; by deriving the equation using other theorems.

39 SN s ENEL 353 Fall 2016 Slide Set 3 slide 39/49 T11 and T11 : Consensus Theorems T11: (B C) + (B D) + (C D) = (B C) + (B D) T11, the dual of T11: (B + C) (B + D) (C + D) = (B + C) (B + D) These allow dropping a redundant product from an SOP expression (T11) or redundant sum from an POS expression (T11 ). Opportunities to use them are not easy to spot when doing algebraic manipulation, but it turns out that T11 justifies simplifications done using Karnaugh maps, as we ll see a little later in the course.

40 SN s ENEL 353 Fall 2016 Slide Set 3 slide 40/49 Do students need to memorize all the numbers and names for theorems? For the most part, the answer is No. You will be tested on algebraic manipulation, but that is much more a matter of finding allowable steps in manipulations than it is a matter of knowing names of theorems. Example: You don t need to know the names and numbers for theorems that allow you to do this... A = A 1 = A(B + B) = AB + AB... but you must know that it is a valid and often useful step. A big exception: It is good to know exactly what is meant by De Morgan s Theorem!

41 SN s ENEL 353 Fall 2016 Slide Set 3 slide 41/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

42 SN s ENEL 353 Fall 2016 Slide Set 3 slide 42/49 T12 and T12 : De Morgan s Theorem Here are T12 and its dual, for N input variables: T12: B 1 B 2... B N = B 1 + B B N T12 : B 1 + B B N = B 1 B 2... B N For the cases of N = 2, N = 3, and N = 4, let s write out De Morgan s Theorem, using input variables A, B, C and D, as necessary.

43 SN s ENEL 353 Fall 2016 Slide Set 3 slide 43/49 Mistakes you must avoid! If you are doing algebraic manipulation just a little faster than you are thinking, you might do things like AB = Ā B (Incorrect!) or A + B + C = A + B + C (Also incorrect!) What are correct transformations for AB and A + B + C?

44 SN s ENEL 353 Fall 2016 Slide Set 3 slide 44/49 De Morgan s Theorem, NAND gates and NOR gates A NAND gate performs a NOT-of-the-AND function. De Morgan s Theorem says that we can also view that as an OR-of-the-NOTs function. So the following two symbols both describe the behaviour of a single circuit element... A B AB A B A + B = AB Let s write a similar description of a 3-input NOR gate and draw two different symbols for the gate.

45 SN s ENEL 353 Fall 2016 Slide Set 3 slide 45/49 De Morgan s Theorem and POS Canonical Form Review... Goal Step 1 Step 2 SOP canonical For truth table rows OR together form with F = 1, find true minterms found in POS canonical form minterms. For truth table rows with F = 0, find false maxterms. Step 1. AND together maxterms found in Step 1. Suppose that the method for finding SOP canonical form makes sense to you, but the method for POS canonical form is a little mysterious. Let s demonstrate how POS canonical form for F can be found starting with SOP canonical form for F.

46 SN s ENEL 353 Fall 2016 Slide Set 3 slide 46/49 Outline of Slide Set 3 Ordinary Algebra Boolean Algebra Order of operations in Boolean algebra Sum-of-products expressions Products of sums and product-of-sums canonical form Axioms and Theorems of Boolean Algebra T12 and T12 : De Morgan s Theorem Using theorems to simplify expressions

47 SN s ENEL 353 Fall 2016 Slide Set 3 slide 47/49 Using theorems to simplify expressions A practical application of Boolean algebra theorems is to find an economical expression for a logic function. Usually, using fewer gates with smaller numbers of inputs per gate leads to circuits that use less chip area; consume less power; switch output faster when inputs change from 0 to 1 or 1 to 0. Example: Canonical SOP for the C OUT function in a 1-bit full adder is ABC IN + ABC IN + ABC IN + ABC IN. Let s use algebraic manipulation to find a simpler SOP expression for C OUT (A,B,C IN ).

48 SN s ENEL 353 Fall 2016 Slide Set 3 slide 48/49 Using theorems to simplify expressions often works well, but it raises a few questions, such as... What is a good choice of theorem to use to make my expression simpler? What is a bad choice, to be avoided, because it would make my expression more complicated, rather than simpler? It s so hard to avoid making mistakes that result in invalid expressions! How the **** do I make sure things don t go wrong?!?! How do I know when I m done how do I know there are no more simplifications to be made?

49 SN s ENEL 353 Fall 2016 Slide Set 3 slide 49/49 Unfortunately, the question on the previous slide don t always have simple answers! We ll see later in the course that Karnaugh maps help with answers to these questions, especially in the cases of 3 5 input variables.

Slides for Lecture 10 Slides for Lecture 10 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 30 September, 2013 ENEL 353

Logic Design. Chapter 2: Introduction to Logic Circuits Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic

CHAPTER III BOOLEAN ALGEBRA CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.

Administrative Notes. Chapter 2 <9> Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2

CHAPTER III BOOLEAN ALGEBRA CHAPTER III- CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III-2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.

Functions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)

CHAPTER1: Digital Logic Circuits Combination Circuits CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.

Ex: Boolean expression for majority function F = A'BC + AB'C + ABC ' + ABC. Boolean Expression Forms: Sum-of-products (SOP) Write an AND term for each input combination that produces a 1 output. Write the input variable if its value is 1; write its complement otherwise. OR the

This form sometimes used in logic circuit, example: Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from

Chapter 2: Switching Algebra and Logic Circuits Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used

Combinational Logic. Review of Combinational Logic 1 Combinational Logic! Switches -> Boolean algebra! Representation of Boolean functions! Logic circuit elements - logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational

ENG2410 Digital Design Combinational Logic Circuits ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single

Unit 2 Session - 6 Combinational Logic Circuits Objectives Unit 2 Session - 6 Combinational Logic Circuits Draw 3- variable and 4- variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the Product-of-Sums

T02 Tutorial Slides for Week 6 T02 Tutorial Slides for Week 6 ENEL 353: Digital Circuits Fall 2017 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 17 October, 2017

ELCT201: DIGITAL LOGIC DESIGN ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter

Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The

Combinational Logic Fundamentals Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic

ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of

Number System conversions Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number

Digital Circuit And Logic Design I. Lecture 3 Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinational-circuit analysis 4. Combinational-circuit synthesis Panupong

EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive EECS150 - Digital Design Lecture 19 - Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150 - Lec19-cl1 Page 1 Boolean Algebra I (Representations of Combinational

Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York

EECS Variable Logic Functions EECS150 Section 1 Introduction to Combinational Logic Fall 2001 2-Variable Logic Functions There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs X

Chapter 2 Combinational logic Chapter 2 Combinational logic Chapter 2 is very easy. I presume you already took discrete mathemtics. The major part of chapter 2 is boolean algebra. II - Combinational Logic Copyright 24, Gaetano Borriello

2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital

Lecture 2 Review on Digital Logic (Part 1) Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active

Week-I. Combinational Logic & Circuits Week-I Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and

211: Computer Architecture Summer 2016 211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic - Storage: Recap - Review: cache hit rate - Project3 - Digital Logic: - truth table => SOP - simplification: Boolean

Possible logic functions of two variables ombinational logic asic logic oolean algebra, proofs by re-writing, proofs by perfect induction logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,..., minimal set Logic realization

ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802

Boolean Algebra and Digital Logic All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types

CS 226: Digital Logic Design CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions

MC9211 Computer Organization MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra

CPE100: Digital Logic Design I Chapter 2 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris Combinational Logic Design Chapter

II. COMBINATIONAL LOGIC DESIGN. - algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT): ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra - developed by George Boole

CS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions

Chapter 2: Princess Sumaya Univ. Computer Engineering Dept. hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra

Standard Expression Forms ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms

EEE130 Digital Electronics I Lecture #4 EEE130 Digital Electronics I Lecture #4 - Boolean Algebra and Logic Simplification - By Dr. Shahrel A. Suandi Topics to be discussed 4-1 Boolean Operations and Expressions 4-2 Laws and Rules of Boolean

Combinational logic. Possible logic functions of two variables. Minimal set of functions. Cost of different logic functions. Combinational logic Possible logic functions of two variables Logic functions, truth tables, and switches NOT, ND, OR, NND, NOR, OR,... Minimal set xioms and theorems of oolean algebra Proofs by re-writing

Combinational Logic Design Principles Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems

Chapter 2: Boolean Algebra and Logic Gates Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a

Ch 2. Combinational Logic. II - Combinational Logic Contemporary Logic Design 1 Ch 2. Combinational Logic II - Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:

Digital Logic Design. Combinational Logic Digital Logic Design Combinational Logic Minterms A product term is a term where literals are ANDed. Example: x y, xz, xyz, A minterm is a product term in which all variables appear exactly once, in normal

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2 - Part 1 2 Chapter 2 - Part 1 3 Chapter 2 - Part 1 4 Chapter 2 - Part

Boolean Algebra & Logic Gates. By : Ali Mustafa Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions

EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) September 5, 2002 John Wawrzynek Fall 2002 EECS150 Lec4-bool1 Page 1, 9/5 9am Outline Review of

Outline. EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) Combinational Logic (CL) Defined EECS150 - Digital Design Lecture 4 - Boolean Algebra I (Representations of Combinational Logic Circuits) January 30, 2003 John Wawrzynek Outline Review of three representations for combinational logic:

Chapter 2 Boolean Algebra and Logic Gates Ch1: Digital Systems and Binary Numbers Ch2: Ch3: Gate-Level Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan

Simplification of Boolean Functions. Dept. of CSE, IEM, Kolkata Simplification of Boolean Functions Dept. of CSE, IEM, Kolkata 1 Simplification of Boolean Functions: An implementation of a Boolean Function requires the use of logic gates. A smaller number of gates,

Midterm1 Review. Jan 24 Armita Midterm1 Review Jan 24 Armita Outline Boolean Algebra Axioms closure, Identity elements, complements, commutativity, distributivity theorems Associativity, Duality, De Morgan, Consensus theorem Shannon

Slides for Lecture 16 Slides for Lecture 6 ENEL 353: Digital Circuits Fall 203 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 6 October, 203 ENEL 353 F3 Section

Chapter 2 Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates Huntington Postulates 1. (a) Closure w.r.t. +. (b) Closure w.r.t.. 2. (a) Identity element 0 w.r.t. +. x + 0 = 0 + x = x. (b) Identity element 1 w.r.t.. x 1 =

EC-121 Digital Logic Design EC-121 Digital Logic Design Lecture 2 [Updated on 02-04-18] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table. The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or

Combinatorial Logic Design Principles Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 4-1 Boolean algebra a.k.a. switching algebra deals with boolean values -- 0, 1 Positive-logic convention analog voltages

Logic Gate Level. Part 2 Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively

Learning Objectives. Boolean Algebra. In this chapter you will learn about: Ref. Page Slide /78 Learning Objectives In this chapter you will learn about: oolean algebra Fundamental concepts and basic laws of oolean algebra oolean function and minimization Logic gates Logic circuits

CHAPTER 3 BOOLEAN ALGEBRA CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 Exclusive-OR and Equivalence Operations 3.3 The Consensus

Boolean Algebra and Logic Gates Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic

Chapter-2 BOOLEAN ALGEBRA Chapter-2 BOOLEAN ALGEBRA Introduction: An algebra that deals with binary number system is called Boolean Algebra. It is very power in designing logic circuits used by the processor of computer system.

Unit 2 Boolean Algebra Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2

SAMPLE ANSWERS MARKER COPY Page 1 of 12 School of Computer Science 60-265-01 Computer Architecture and Digital Design Fall 2012 Midterm Examination # 1 Tuesday, October 23, 2012 SAMPLE ANSWERS MARKER COPY Duration of examination:

Signals and Systems Digital Logic System Signals and Systems Digital Logic System Prof. Wonhee Kim Chapter 2 Design Process for Combinational Systems Step 1: Represent each of the inputs and outputs in binary Step 1.5: If necessary, break the

UNIVERSITI TENAGA NASIONAL. College of Information Technology UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours

Chap 2. Combinational Logic Circuits Overview 2 Chap 2. Combinational Logic Circuits Spring 24 Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization Two-Level Optimization

Chapter 2 (Lect 2) Canonical and Standard Forms. Standard Form. Other Logic Operators Logic Gates. Sum of Minterms Product of Maxterms Chapter 2 (Lect 2) Canonical and Standard Forms Sum of Minterms Product of Maxterms Standard Form Sum of products Product of sums Other Logic Operators Logic Gates Basic and Multiple Inputs Positive and

Chapter 2 : Boolean Algebra and Logic Gates Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 1431-1432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean

Slide Set 6. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary Slide Set 6 for ENEL 353 Fall 2017 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2017 SN s ENEL 353 Fall 2017 Slide Set 6 slide

ECE 545 Digital System Design with VHDL Lecture 1A. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture A Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Laws

1 Boolean Algebra Simplification cs281: Computer Organization Lab3 Prelab Our objective in this prelab is to lay the groundwork for simplifying boolean expressions in order to minimize the complexity of the resultant digital logic circuit.

Chapter 7 Logic Circuits Chapter 7 Logic Circuits Goal. Advantages of digital technology compared to analog technology. 2. Terminology of Digital Circuits. 3. Convert Numbers between Decimal, Binary and Other forms. 5. Binary

CHAPTER 12 Boolean Algebra 318 Chapter 12 Boolean Algebra CHAPTER 12 Boolean Algebra SECTION 12.1 Boolean Functions 2. a) Since x 1 = x, the only solution is x = 0. b) Since 0 + 0 = 0 and 1 + 1 = 1, the only solution is x = 0. c)

Switches: basic element of physical implementations Combinational logic Switches Basic logic and truth tables Logic functions Boolean algebra Proofs by re-writing and by perfect induction Winter 200 CSE370 - II - Boolean Algebra Switches: basic element

ECE 238L Boolean Algebra - Part I ECE 238L Boolean Algebra - Part I August 29, 2008 Typeset by FoilTEX Understand basic Boolean Algebra Boolean Algebra Objectives Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand

Total Time = 90 Minutes, Total Marks = 50. Total /50 /10 /18 University of Waterloo Department of Electrical & Computer Engineering E&CE 223 Digital Circuits and Systems Midterm Examination Instructor: M. Sachdev October 23rd, 2007 Total Time = 90 Minutes, Total

CHAPTER 2 BOOLEAN ALGEBRA CHAPTER 2 BOOLEAN ALGEBRA This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operations 2.3 Boolean Expressions and Truth Tables 2.4 Basic Theorems 2.5 Commutative, Associative,

Binary Logic and Gates. Our objective is to learn how to design digital circuits. Binary Logic and Gates Introduction Our objective is to learn how to design digital circuits. These circuits use binary systems. Signals in such binary systems may represent only one of 2 possible values

Discrete Mathematics. CS204: Spring, Jong C. Park Computer Science Department KAIST Discrete Mathematics CS204: Spring, 2008 Jong C. Park Computer Science Department KAIST Today s Topics Combinatorial Circuits Properties of Combinatorial Circuits Boolean Algebras Boolean Functions and

Chapter 2. Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common

Logic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 1 Gate Circuits and Boolean Equations Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hperlinks are active in

Boolean Algebra. The Building Blocks of Digital Logic Design. Section. Section Overview. Binary Operations and Their Representation. Section 3 Boolean Algebra The Building Blocks of Digital Logic Design Section Overview Binary Operations (AND, OR, NOT), Basic laws, Proof by Perfect Induction, De Morgan s Theorem, Canonical and Standard

EE40 Lec 15. Logic Synthesis and Sequential Logic Circuits EE40 Lec 15 Logic Synthesis and Sequential Logic Circuits Prof. Nathan Cheung 10/20/2009 Reading: Hambley Chapters 7.4-7.6 Karnaugh Maps: Read following before reading textbook http://www.facstaff.bucknell.edu/mastascu/elessonshtml/logic/logic3.html

Slides for Lecture 19 Slides for Lecture 19 ENEL 353: Digital Circuits Fall 2013 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 23 October, 2013 ENEL 353

Review for Test 1 : Ch1 5 Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8

Every time has a value associated with it, not just some times. A variable can take on any value within a range Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»

Chapter 2 Boolean Algebra and Logic Gates Chapter 2 Boolean Algebra and Logic Gates The most common postulates used to formulate various algebraic structures are: 1. Closure. N={1,2,3,4 }, for any a,b N we obtain a unique c N by the operation

BOOLEAN ALGEBRA TRUTH TABLE BOOLEAN ALGEBRA TRUTH TABLE Truth table is a table which represents all the possible values of logical variables / statements along with all the possible results of the given combinations of values. Eg:

Fundamentals of Computer Systems Fundamentals of Computer Systems Boolean Logic Stephen A. Edwards Columbia University Fall 2011 Boolean Logic George Boole 1815 1864 Boole s Intuition Behind Boolean Logic Variables x, y,... represent

Minimization techniques Pune Vidyarthi Griha s COLLEGE OF ENGINEERING, NSIK - 4 Minimization techniques By Prof. nand N. Gharu ssistant Professor Computer Department Combinational Logic Circuits Introduction Standard representation

S C F F F T T F T T S C B F F F F F T F T F F T T T F F T F T T T F T T T EECS 270, Winter 2017, Lecture 1 Page 1 of 6 Use pencil! Say we live in the rather black and white world where things (variables) are either true (T) or false (F). So if S is Mark is going to the Store

Chapter 2 Combinational Logic Circuits Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal

L2: Combinational Logic Design (Construction and Boolean Algebra) L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson

ELC224C. Karnaugh Maps KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to K-maps How to use K-maps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on 1 COE 202- Digital Logic Binary Logic and Gates Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Introduction Boolean Algebra Elements of Boolean Algebra (Binary Logic) Logic Operations & Logic E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean