...applications to the design of computing machines, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science. A proposition is a statement that is either true or false, but not both. Letters are used to denote propositions, just as letters are used to denote variables. The conventional letters used for this purpose are p, q, r, s, … The truth value of a proposition is true, denoted by T, if it is a true proposition and false, denoted by F, if it is a false proposition. We now turn our attention to methods for producing new propositions from those that we already have. Many mathematical statements are constructed by combining one or more propositions. New propositions, called compound propositions, are formed from existing propositions using logical operators. Let p be a proposition. The statement “It is not the case that p” is another proposition, called the negation of p. The negation of p is denoted by p. The proposition p is read “not p”. A truth table displays the relationships between the truth values of propositions. Table 1. The truth table for the negation of a proposition | P | p | TF | FT | The negation of a proposition can also be considered the result of the operation of the negation operator on a proposition. The negation operator constructs a new proposition from a single existing proposition. We will now introduce the logical operators that are used to form new propositions from two or more existing...
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...Vol 3 Issue 2 March 2013 Impact Factor : 0.2105 ORIGINAL ARTICLE ISSN No : 2230-7850 Monthly Multidisciplinary Research Journal Indian Streams Research Journal Executive Editor Ashok Yakkaldevi Editor-in-chief H.N.Jagtap IMPACT FACTOR : 0.2105 Welcome to ISRJ RNI MAHMUL/2011/38595 ISSN No.2230-7850 Indian Streams Research Journal is a multidisciplinary research journal, published monthly in English, Hindi & Marathi Language. All research papers submitted to the journal will be double - blind peer reviewed referred by members of the editorial Board readers will include investigator in universities, research institutes government and industry with research interest in the general subjects. International Advisory Board Flávio de São Pedro Filho Federal University of Rondonia, Brazil Hasan Baktir Mohammad Hailat English Language and Literature Dept. of Mathmatical Sciences, University of South Carolina Aiken, Aiken SC Department, Kayseri Kamani Perera 29801 Regional Centre For Strategic Studies, Sri Ghayoor Abbas Chotana Lanka Department of Chemistry, Lahore Abdullah Sabbagh University of Management Sciences [ PK Engineering Studies, Sydney Janaki Sinnasamy ] Librarian, University of Malaya [ Anna Maria Constantinovici Catalina Neculai Malaysia ] AL. I. Cuza University, Romania University of Coventry, UK Romona Mihaila Spiru Haret University, Romania Delia Serbescu Spiru Haret University, Bucharest, Romania Anurag Misra DBS College, Kanpur Titus Pop Ecaterina...
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...ENGI 241 Experiment 5 Basic Logic Gates OBJECTIVE This experiment will examine the operation of the AND, NAND, OR, and NOR logic gates and compare the expected outputs to the truth tables for these devices. The NOT function will be implemented using NAND and NOR gates. EQUIPMENT AND PARTS REQUIRED 1 Protoboard 1 DVM 1 Logic Probe 1 74LS00, 74LS02, 74LS08, 74LS32 2 each 1kΩ resistors INTRODUCTION The NOT circuit or inverter performs the basic logic function of complementation. It may be identified by the presence of a bubble on the input or the output of the traditional logic symbol or a triangle on the IEEE/IEC logic symbol as seen in Figure 1. The inverter has one input and one output and the output is the complement of the input. Figure 1 contains the truth table for the NOT function. LOGIC DIAGRAM TRUTH TABLE A Q 0 1 1 0 FIGURE 1 All logic gates have two or more inputs and one output. These logic gates accept digital logic levels on their inputs and will provide a digital logic level output which is dependent on the type of logic gate and the inputs applied to the gate. For the TTL logic family, any gate input that is not connected will be treated as if a logic 1 is present on that input. The number of different possible combinations of inputs is 2n where n is the number of inputs. Therefore, four unique combinations of inputs are possible for a two input gate. A B Q 0 0 1 1 FIGURE 2 0 1 0 1 0 0 0 1 The AND function is similar to the multiplication in mathematics...
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...Math 221 Notes Basic Connectives and Truth Tables In the development of any mathematical theory, assertions are made in the form of sentences. Such verbal or written assertions, called statements (or propositions), are declarative sentences that are either true or false—but not both. For example, the following are statements, and we use the lowercase letters of the alphabet (such as p, q, and r) to represent these statements. p: Combinatorics is a required course for sophomores. q: Margaret Mitchell wrote Gone with the Wind. r: 2 + 3 _ 5. The preceding statements represented by the letters p, q, and r are considered to be primitive statements, for there is really no way to break them down into anything simpler. New statements can be obtained from existing ones in two ways. 1) Transform a given statement p into the statement ￢p, which denotes its negation and is read “Not p.” For the statement p above, ￢p is the statement “Combinatorics is not a required course for sophomores.” (We do not consider the negation of a primitive statement to be a primitive statement.) 2) Combine two or more statements into a compound statement, using the following logical connectives. a) Conjunction: The conjunction of the statements p, q is denoted by p ∧ q, which is read “p and q.” In our example the compound statement p ∧ q is read “Combinatorics is a required course for sophomores, and Margaret Mitchell wrote Gone with the Wind.” b) Disjunction: The expression p ∨ q denotes...
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...Computer Science Chapter 1: Introduction to Computer Hardware Different Categories of Computer and Computing Devices Tablets The lightest and most portable Touch interface, good for “light” work Laptops/Notebooks Larger display area; adds CD or DVD as well as physical keyboard They are portable; price for performance is not as good as desktop, choice of hardware is limited Specialized Variant Laptops Ultrabooks Thinner, and lighter than laptops Cost is higher than laptop (all hardware being equal) Netbooks Cheaper more portable laptop that is smaller and has a lower quality display and overall less powerful hardware Much less common than tablets today Desktop Computers Everything is separate (monitor, computer, keyboard, etc); this allows you to mix and match and customize your desktop computer, at the cost of increased complexity (some compatibility issues may arise – not everything works together) and decreased portability. Larger ‘footprint’ (More space is required, but this allows for increase expandability) ------------------------------------------------- Reduced costs/more options (compared to laptops) ------------------------------------------------- The purpose of an operating system is to run the computer. The operating system determines the interface of a computer, its configurability, and its security. In general, due to popularity and tweak-ability, the MS-WINDOWS (PC) OS has more viruses than the MAC OS. In general, the MAC OS...
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...15.2.2016 Reduction SAT Problem Group 1 1 Reduction 2 Reduction an important concept for understanding the relationship between problems. 3 solving one problem in terms of another Example : Suppose you have some problem A that you don’t know how to solve. If you can find a way to reduce problem A to some problem B that you do know how to solve, then that’s just as good as finding a way to solve A in the first place. 4 SORTING: Input: A sequence of integers x0, x1, x2, ..., xn−1. Output: A permutation y0, y1, y2, ..., yn−1 of the sequence such that yi ≤ yj whenever i < j. PAIRING: Input: Two sequences of integers X = (x0, x1, ..., xn−1) and Y = (y0, y1, ..., yn−1). Output: A pairing of the elements in the two sequences such that the least value in X is paired with the least value in Y, the next least value in X is paired with the next least value in Y, and so on. 5 An illustration of PAIRING. The two lists of numbers are paired up so that the least values from each list make a pair, the next smallest values from each list make a pair, and so on. 6 Solution PAIRING is to use an existing sorting program to sort each of the two sequences, and then pair off items based on their position in sorted order. PAIRING is reduced to SORTING, because SORTING is used to solve PAIRING. 7 3-step Process 1. convert an instance of PAIRING into two instances of SORTING . 2. sort the two arrays . 3. convert...
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...CS 331 – Introduction to Artificial Intelligence Assignment 2 Solution 1. A new operator ⊕, or exclusive-or, may be defined by the following truth table: P T T F F P⊕Q F T T F Q T F T F Create a propositional calculus expression using only ^, v, and ~ that is equivalent to P ⊕ Q. Prove their equivalence using truth tables. One possible correct answer is: P ⊕ Q ≡ (P v Q) ^ ~(P ^ Q) Proof using truth tables: P T T F F Q T F T F (P v Q) T T T F (P ^ Q) T F F F ~(P ^ Q) F T T T (P v Q) ^ ~(P ^ Q) F T T F P⊕Q F T T F 2. The logical operator “ ” is read “if and only if.” P Q is defined as being equivalent to (P Q) ^ (Q P). Based on this definition, show that P Q is logically equivalent to (P v Q) (P ^ Q) using truth tables. Truth Table for (P P T T F F Q) ^ (Q Q T F T F Truth Table for (P v Q) P T T F F Q T F T F P) P -> Q T F T T Q -> P T T F T (P Q) ^ (Q P^Q T F F F (P v Q) P) T F F T (P ^ Q) PvQ T T T F (P ^ Q) T F F T Hence shown that both expressions are logically equivalent. 3. Assuming this is a game in which animals attack each other, try to represent the following situation completely using first-order logic: You have to represent the following facts: a. The location (air/ground) of the animals b. The abilities of the animals (e.g. flying) c. The relative speed of the animals (slow/fast) d. Line-of-sight status (animal A is visible to animal B) Answers can be moderately different from the following suggested answer: is_at_location(turtle, ground)...
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...connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic. Semantics of a logical connective is often, but not always, presented as a truth function. A logical connective is similar to but not equivalent to a conditional operator. [1] Contents [hide] 1 In language 1.1 Natural language 1.2 Formal languages 2 Common logical connectives 2.1 List of common logical connectives 2.2 History of notations 2.3 Redundancy 3 Properties 4 Order of precedence 5 Computer science 6 See also 7 Notes 8 References 9 Further reading 10 External links In language[edit] Natural language[edit] In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a grammatically compound sentence. Some but not all such grammatical conjunctions are truth functions. For example, consider the following sentences: A: Jack went up the hill. B: Jill went up the hill. C: Jack went up the hill and Jill went up the hill. D: Jack went up the hill so Jill went up the hill. The words and and so are grammatical conjunctions joining the sentences (A) and (B) to form the compound sentences (C) and (D). The and in (C) is a logical connective, since the truth of (C) is completely determined by (A) and (B): it would make no sense...
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...Final Exam Review 1. Topics to study a) Chapter 2 – Fundamentals of Logic b) Chapter 3 – Set Theory c) Chapter 4 – Induction and the Division Algorithm d) Chapter 5 & 7 – Relations and Functions e) Chapter 1 – Combinatorics f) Chapter 8 – Inclusion/Exclusion Principle 2. Chapter 2 – Fundamentals of Logic a) Logical Connectives i) Conjunction ii) Disjunction iii) Negation iv) XOR v) Implication vi) Bi-Direction b) Truth Table construction 8. Construct a truth table for each of the following compound statements, where p, q, r denote primitive statements. Tell whether the given statement is a tautology. a. ¬ (p Ú ¬q) → ¬p p q ¬q ¬p p Ú ¬q ¬ (p Ú ¬q) ¬ (p Ú ¬q) → ¬p 0 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 0 1 The statement is a tautology. b. p → (q → r) p q r q → r p → (q → r) 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 The statement is not a tautology. c. (p → q) → r p q r p → q (p → q) → r 0 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 The statement is not a tautology. d. (p → q) → (q → p) p q p → q q → p (p → q) → (q → p) 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 The statement is not a tautology. e. (p (p → q)) → q p q p → q p (p → q) (p (p → q)) → q 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1 1 The statement is a tautology. f. (p q) → p p q p q (p q) → p 0 0 0 1 0 1 0 1 1...
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...Dear granny. As you know, I’m taking the Digital electronics class at my school. We are doing this project where we create a system to calculate majority vote. with the system we used variables to take the place as the president, vice president, secretary, and treasurer. the way we started this was we had to use binary where zero is a no vote, while a one is a yes vote. whenever it would be a tie, the president's vote would be the tie breaker. so if the vice president, and the secretary voted yes, but the president and treasurer voted no, since the president's vote is a no, the majority vote would end with a no vote. we had to create a truth table of every different outcome to make sure everything would check out. then after that we would get all of the ones with a positive outcome, and then write them all out as if it were a math problem. then using this stuff called boolean algebra, we had to make the bigger math problem much simpler so it would fit into the circuit system that we use to simulate how it would look in real life. the simplification was very long and quite difficult. it is very confusing and very difficult to explain. so to simplify it, the example i'm going to give you is very simple compared to what we had to do. so say that the president said no, so in the simplification, it would be not P, or not president. and continuing this same process down the line until you would get them all. say that two terms would be PT and the others would be P and not T, according...
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...AT-700 and check properly the connections. Remember to connect each IC’s pin 14 to “+5V” of DC power supply of AT-700 and pin 7 to “GND”. Step 3: Connect the Data switches “0” and “1” to point A and B of fig. 1-1 respectively. Then connect 8 bit LED Display’s “0”, “1” and “2” to the output of point Y1, Y2 and Y3 of Fig. 1-1 respectively. The connection diagram is as follows. [pic] Step 4: Change Data Switches “0” and “1” between “0” and “1” position and observe the situation of 8 bit LED Display “0”, “1” and “2”. The LED is light that indicates the output is in the logic 1 condition. When LED is dark, it indicates that the output is in the logic 0 condition. Step 5: Record the results that you have observed into the truth table as follows. Truth Table 1-3 [pic] Discussion & Conclusion: (On the basis of experimental...
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...Lecture # 1 – Concepts in Programming System Development Life Cycle (SDLC) - Is a series of well-defined steps that should be followed when a system is created or changed. - Represents the big picture of what happens during system creation or modification. The System Development Life Cycle is comprised of 6 steps, namely: 1. Analyze the current system 2. Define the new system requirements 3. Design the new system 4. Develop the new system 5. Implement the new system 6. Evaluate the new system The Program Development Cycle (PDC) - The fourth step of the SDLC (Development of the new system) is comprised of a series of well-defined steps called the Program Development Cycle. - A programmer carries out the steps in the PDC. The steps are outlined as follows: 1. Review the input, processing, output and storage requirements 2. Develop the logic for the program 3. Write the program using a programming language 4. Test and debug the program 5. Complete the program documentation What is a Program? - A program is an organized lists of instructions that, when executed, causes the computer to behave in a predetermined manner. Without programs, computers are useless. What is Programming? - Programming is instructing a computer to do something for you with the help of a programming language Identifiers Manipulation of data is a basic feature of a computer’s abilities. This...
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...Logical Operations and Truth Tables At first glance, it may not seem that the study of logic should be part of mathematics. For most of us, the word logic is associated with reasoning in a very nebulous way: "If my car is out of gas, then I cannot drive it to work." seems logical enough, while "If I am curious, then I am yellow." is clearly illogical. Yet our conclusions about what is or is not logical are most often unstructured and subjective. The purpose of logic is to enable the logician to construct valid arguments which satisfy the basic principle "If all of the premises are true, then the conclusion must be true." It turns out that in order to reliably and objectively construct valid arguments, the logical operations which one uses must be clearly defined and must obey a set of consistent properties. Thus logic is quite rightly treated as a mathematical subject. Up until now, you've probably considered mathematics as a set of rules for using numbers. The study of logic as a branch of mathematics will require you to think more abstractly then you are perhaps used to doing. For instance, in logic we use variables to represent propositions (or premises), in the same fashion that we use variables to represent numbers in algebra. But while an algebraic variable can have any number as its value, a logical variable can only have the value True or False. That is, True and False are the "numerical constants" of logic. And instead of the usual arithmetic operators (addition...
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...Week 5 Individual Assignment Chapter 15 Supplementary Exercises, problems 1, 5, & 6 1. Let n ≥ 2. If xi is a Boolean variable for all 1 ≤ i ≤ n, prove that a) (x1 + x2 + ・ ・ ・ + xn) _ x1x2 ・ ・ ・ xn Assume the result for n _ k (≥ 2) and consider the case of n _ k + 1. b) (x1x2 ・ ・ ・ xn) _ x1 + x2 + ・ ・ ・ + xn Follows from part (a) by duality. 5. Let_be a Boolean algebra that is partially ordered by≤. If x, y, z ∈ _, prove that x + y ≤ z if and only if x ≤ z and y ≤ z. If x ≤ z and y ≤ z, then from Exercise 6(b) of Section 15.4 we have x + y ≤ z + z. z + z _ z. Conversely, suppose that x + y ≤ z.We find that x ≤ x + y, because x(x + y) _ x + xy _ x. Since x ≤ x + y and x + y ≤ z, we have x ≤ z, because a partial order is transitive. 6. State and prove the dual of the result in Exercise 5. Exercise 15.1, problems 1, 2, 11, 12, 14, & 15 1. Find the value of each of the following Boolean expressions if the values of the Boolean variables w, x, y, and z are 1, 1, 0, and 0, respectively. a) xy + x y = 1 b) w + xy = 1 c) wx + y + yz = 1 d) (wx + yz) + wy + (w + y)(x + y) = 1 2. Let w, x, and y be Boolean variables where the value of x is 1. For each of the following Boolean expressions, determine, if possible, the value of the expression. If you cannot determine the value of the expression, then find the number of assignments of values for w and y that will result in the value 1 for the expression. a) x + xy + w...
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...TRUE and FALSE. Some logical functions return a Boolean value as their result, others use the Boolean result of a comparison to choose between alternative calculations. There are six functions listed in the logical group in Excel 2003 – the functions AND, FALSE, IF, NOT, OR, TRUE – and a seventh in Excel 2007 – the function IFERROR. You’ll see the use of most of these in this lab. First, however, it’s worthwhile to become familiar with the logical operators. Logical Operators TRUE and FALSE are common concepts. They are values which pertain to statements. For example, the statement “It is morning.” is either TRUE or FALSE. We recognize that its truth value may change, but at any particular time the statement is either TRUE or FALSE. What may be hidden here is the existence of an implied comparison. To determine the truth value of any statement we compare our understanding of the meaning of the claim with the facts. Strictly speaking the statement “It is morning.” means the time of day is after midnight and before noon. To decide if it’s TRUE we need to know the actual time of day and compare it to our criteria. It’s in these comparisons that we use Logical Operators: |Comparison |Symbol | |less than |< | |less than or equal to |= | |greater than |> | |less than or greater than...
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