Forall X in Lurch
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
1 Elementary Set Theory
1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection). -
Application: Digital Logic Circuits
SECTION 2.4 Application: Digital Logic Circuits Copyright © Cengage Learning. All rights reserved. Application: Digital Logic Circuits Switches “in series” Switches “in parallel” Change closed and on are replaced by T, open and off are replaced by F? Application: Digital Logic Circuits • More complicated circuits correspond to more complicated logical expressions. • This correspondence has been used extensively in design and study of circuits. • Electrical engineers use language of logic when refer to values of signals produced by an electronic switch as being “true” or “false.” • Only that symbols 1 and 0 are used • symbols 0 and 1 are called bits, short for binary digits. • This terminology was introduced in 1946 by the statistician John Tukey. Black Boxes and Gates Black Boxes and Gates • Circuits: transform combinations of signal bits (1’s and 0’s) into other combinations of signal bits (1’s and 0’s). • Computer engineers and digital system designers treat basic circuits as black boxes. • Ignore inside of a black box (detailed implementation of circuit) • focused on the relation between the input and the output signals. • Operation of a black box is completely specified by constructing an input/output table that lists all its possible input signals together with their corresponding output signals. Black Boxes and Gates One possible correspondence of input to output signals is as follows: An Input/Output Table Black Boxes and Gates An efficient method for designing more complicated circuits is to build them by connecting less complicated black box circuits. Gates can be combined into circuits in a variety of ways. If the rules shown on the next page are obeyed, the result is a combinational circuit, one whose output at any time is determined entirely by its input at that time without regard to previous inputs. -
Shalack V.I. Semiotic Foundations of Logic
Semiotic foundations of logic Vladimir I. Shalack abstract. The article offers a look at the combinatorial logic as the logic of signs operating in the most general sense. For this it is proposed slightly reformulate it in terms of introducing and replacement of the definitions. Keywords: combinatory logic, semiotics, definition, logic founda- tions 1 Language selection Let’s imagine for a moment what would be like the classical logic, if we had not studied it in the language of negation, conjunction, dis- junction and implication, but in the language of the Sheffer stroke. I recall that it can be defined with help of negation and conjunction as follows A j B =Df :(A ^ B): In turn, all connectives can be defined with help of the Sheffer stroke in following manner :A =Df (A j A) A ^ B =Df (A j B) j (A j B) A _ B =Df (A j A) j (B j B) A ⊃ B =Df A j (B j B): Modus ponens rule takes the following form A; A j (B j B) . B 226 Vladimir I. Shalack We can go further and following the ideas of M. Sch¨onfinkel to define two-argument infix quantifier ‘jx’ x A j B =Df 8x(A j B): Now we can use it to define Sheffer stroke and quantifiers. x A j B =Df A j B where the variable x is not free in the formulas A and B; y x y 8xA =Df (A j A) j (A j A) where the variable y is not free in the formula A; x y x 9xA =Df (A j A) j (A j A) where the variable y is not free in the formula A. -
CS 173: Discrete Structures, Spring 2012 Homework 1 Solutions
CS 173: Discrete Structures, Spring 2012 Homework 1 Solutions This homework contains 3 problems worth a total of 25 points. 1. [1 point] Finding information on Piazza How many pet rats does Margaret’s family have? (Hint: see Piazza.) Solution: Margaret’s family has six pet rats. 2. [4 points] Demographic Survey 3. [20 points] Logic operators The late 19th century philosopher Charles Peirce (rhymes with ‘hearse,’ not ‘fierce’) wrote about a set of logically dual operators and, in his writings, coined the term ‘Ampheck’ to describe them. The two most common Ampheck operators, the Peirce arrow (written ↓ or ⊥ or ∨ by different people) and the Sheffer stroke (written ↑ or | or ∧ by different people), are defined by the following truth table: p q p ↑ q p ↓ q T T F F T F T F F T T F F F T T (a) (4 points) The set of operators {∧, ∨, ¬} is functionally complete, which means that every logical statement can be expressed using only these three operators. Is the smaller set of operators {∨, ¬} also functionally complete? Explain why or why not. Solution: By De Morgan’s laws, ¬(¬p∨¬q) ≡ ¬¬p∧¬¬q ≡ p∧q So every logical statement using the ∧ operator can be rewritten in terms of the ∨ and ¬ operators. Since every logical statement can be expressed in terms of the ∧, ∨, and ¬ operators, this implies that every logical statement can be expressed in terms of the ∨ and ¬ operators, and so {∨, ¬} is functionally complete. (b) (4 points) Express ¬p using only the Sheffer stroke operation ↑. Solution: Note that ¬p is true if and only if p is false. -
Boolean Logic
Boolean logic Lecture 12 Contents . Propositions . Logical connectives and truth tables . Compound propositions . Disjunctive normal form (DNF) . Logical equivalence . Laws of logic . Predicate logic . Post's Functional Completeness Theorem Propositions . A proposition is a statement that is either true or false. Whichever of these (true or false) is the case is called the truth value of the proposition. ‘Canberra is the capital of Australia’ ‘There are 8 day in a week.’ . The first and third of these propositions are true, and the second and fourth are false. The following sentences are not propositions: ‘Where are you going?’ ‘Come here.’ ‘This sentence is false.’ Propositions . Propositions are conventionally symbolized using the letters Any of these may be used to symbolize specific propositions, e.g. :, Manchester, , … . is in Scotland, : Mammoths are extinct. The previous propositions are simple propositions since they make only a single statement. Logical connectives and truth tables . Simple propositions can be combined to form more complicated propositions called compound propositions. .The devices which are used to link pairs of propositions are called logical connectives and the truth value of any compound proposition is completely determined by the truth values of its component simple propositions, and the particular connective, or connectives, used to link them. ‘If Brian and Angela are not both happy, then either Brian is not happy or Angela is not happy.’ .The sentence about Brian and Angela is an example of a compound proposition. It is built up from the atomic propositions ‘Brian is happy’ and ‘Angela is happy’ using the words and, or, not and if-then. -
Logic, Proofs
CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. 1.1.1. Connectives, Truth Tables. Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation p “not p” Conjunction p¬ q “p and q” Disjunction p ∧ q “p or q (or both)” Exclusive Or p ∨ q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p → q “p if and only if q” ↔ The truth value of a compound proposition depends only on the value of its components. Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the following way: 6 1.1. PROPOSITIONS 7 p q p p q p q p q p q p q T T ¬F T∧ T∨ ⊕F →T ↔T T F F F T T F F F T T F T T T F F F T F F F T T Note that represents a non-exclusive or, i.e., p q is true when any of p, q is true∨ and also when both are true. -
Logic, Sets, and Proofs David A
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical statements with capital letters A; B. Logical statements be combined to form new logical statements as follows: Name Notation Conjunction A and B Disjunction A or B Negation not A :A Implication A implies B if A, then B A ) B Equivalence A if and only if B A , B Here are some examples of conjunction, disjunction and negation: x > 1 and x < 3: This is true when x is in the open interval (1; 3). x > 1 or x < 3: This is true for all real numbers x. :(x > 1): This is the same as x ≤ 1. Here are two logical statements that are true: x > 4 ) x > 2. x2 = 1 , (x = 1 or x = −1). Note that \x = 1 or x = −1" is usually written x = ±1. Converses, Contrapositives, and Tautologies. We begin with converses and contrapositives: • The converse of \A implies B" is \B implies A". • The contrapositive of \A implies B" is \:B implies :A" Thus the statement \x > 4 ) x > 2" has: • Converse: x > 2 ) x > 4. • Contrapositive: x ≤ 2 ) x ≤ 4. 1 Some logical statements are guaranteed to always be true. These are tautologies. Here are two tautologies that involve converses and contrapositives: • (A if and only if B) , ((A implies B) and (B implies A)). In other words, A and B are equivalent exactly when both A ) B and its converse are true. -
Lecture 1: Propositional Logic
Lecture 1: Propositional Logic Syntax Semantics Truth tables Implications and Equivalences Valid and Invalid arguments Normal forms Davis-Putnam Algorithm 1 Atomic propositions and logical connectives An atomic proposition is a statement or assertion that must be true or false. Examples of atomic propositions are: “5 is a prime” and “program terminates”. Propositional formulas are constructed from atomic propositions by using logical connectives. Connectives false true not and or conditional (implies) biconditional (equivalent) A typical propositional formula is The truth value of a propositional formula can be calculated from the truth values of the atomic propositions it contains. 2 Well-formed propositional formulas The well-formed formulas of propositional logic are obtained by using the construction rules below: An atomic proposition is a well-formed formula. If is a well-formed formula, then so is . If and are well-formed formulas, then so are , , , and . If is a well-formed formula, then so is . Alternatively, can use Backus-Naur Form (BNF) : formula ::= Atomic Proposition formula formula formula formula formula formula formula formula formula formula 3 Truth functions The truth of a propositional formula is a function of the truth values of the atomic propositions it contains. A truth assignment is a mapping that associates a truth value with each of the atomic propositions . Let be a truth assignment for . If we identify with false and with true, we can easily determine the truth value of under . The other logical connectives can be handled in a similar manner. Truth functions are sometimes called Boolean functions. 4 Truth tables for basic logical connectives A truth table shows whether a propositional formula is true or false for each possible truth assignment. -
Propositional Logic
Mathematical Logic (Based on lecture slides by Stan Burris) George Voutsadakis1 1Mathematics and Computer Science Lake Superior State University LSSU Math 300 George Voutsadakis (LSSU) Logic January 2013 1 / 86 Outline 1 Propositional Logic Connectives, Formulas and Truth Tables Equivalences, Tautologies and Contradictions Substitution Replacement Adequate Sets of Connectives Disjunctive and Conjunctive Forms Valid Arguments, Tautologies and Satisfiability Compactness Epilogue: Other Propositional Logics George Voutsadakis (LSSU) Logic January 2013 2 / 86 Propositional Logic Connectives, Formulas and Truth Tables Subsection 1 Connectives, Formulas and Truth Tables George Voutsadakis (LSSU) Logic January 2013 3 / 86 Propositional Logic Connectives, Formulas and Truth Tables The Alphabet: Connectives and Variables The following are the basic logical connectives that we use to connect logical statements: Symbol Name Symbol Name 1 true ∧ and 0 false ∨ or ¬ not → implies ↔ iff In the same way that in algebra we use x, y, z,... to stand for unknown or varying numbers, in logic we use the propositional variables P, Q, R,... to stand for unknown or varying propositions or statements; Using the connectives and variables we can construct propositional formulas like ((P → (Q ∨ R)) ∧ ((¬Q) ↔ (1 ∨ P))). George Voutsadakis (LSSU) Logic January 2013 4 / 86 Propositional Logic Connectives, Formulas and Truth Tables Inductive (Recursive) Definition of Propositional Formulas Propositional formulas are formally built as follows: Every propositional variable P is a propositional formula; the constants 0 and 1 are propositional formulas; if F is a propositional formula, then (¬F ) is a propositional formula; if F and G are propositional formulas, then (F ∧ G), (F ∨ G), (F → G) and (F ↔ G) are propositional formulas. -
Combinational Circuits
Combinational Circuits Jason Filippou CMSC250 @ UMCP 06-02-2016 Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 1 / 1 Outline Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 2 / 1 Hardware design levels Hardware design levels Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 3 / 1 Hardware design levels Levels of abstraction for ICs Small Scale Integration: ≈ 10 boolean gates Medium Scale Integration: > 10; ≤ 100 Large Scale Integration: Anywhere between 100 and 30; 000. Very Large Scale Integration: Up to 150; 000. Very Very Large Scale Integration: > 150; 000. Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 4 / 1 Hardware design levels Example schematic Figure 1: VLSI schematic for a professional USB interface for Macintosh PCs. Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 5 / 1 Hardware design levels SSI Consists of circuits that contain about 10 gates. 16-bit adders. Encoders / Decoders. MUX/DEMUX. ::: All our examples will be at an SSI level. Logic Design is the branch of Computer Science that essentially analyzes the kinds of circuits and optimizations done at an SSI/MSI level. We do not have such a course in the curriculum, but you can expect to be exposed to it if you ever take 411. Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 6 / 1 Combinational Circuit Design Combinational Circuit Design Jason Filippou (CMSC250 @ UMCP) Circuits 06-02-2016 7 / 1 A sequential circuit is one where this constraint can be violated. Example: Flip-flops (yes, seriously). In this course, we will only touch upon combinational circuits. Combinational Circuit Design Combinational vs Sequential circuits A combinational circuit is one where the output of a gate is never used as an input into it. -
Logic and Bit Operations
Logic and bit operations • Computers represent information by bits. • A bit has two possible values, namely zero and one. This meaning of the word comes from binary digit, since zeroes and ones are the digits used in binary representations of numbers. The well-known statistician John Tuley introduced this terminology in 1946. (There were several other suggested words for a binary digit, including binit and bigit, that never were widely accepted.) • A bit can be used to represent a truth value, since there are two truth values, true and false. • A variable is called a Boolean variable if its values are either true or false. • Computer bit operations correspond to the logical connectives. We will also use the notation OR, AND and XOR for _ , ^ and exclusive _. • A bit string is a sequence of zero or more bits. The length of the string is the number of bits in the string. 1 • We can extend bit operations to bit strings. We define bitwise OR, bitwise AND and bitwise XOR of two strings of the same length to be the strings that have as their bits the OR, AND and XOR of the corresponding bits in the two strings. • Example: Find the bitwise OR, bitwise AND and bitwise XOR of the bit strings 01101 10110 11000 11101 Solution: The bitwise OR is 11101 11111 The bitwise AND is 01000 10100 and the bitwise XOR is 10101 01011 2 Boolean algebra • The circuits in computers and other electronic devices have inputs, each of which is either a 0 or a 1, and produce outputs that are also 0s and 1s. -
Proofs and Mathematical Reasoning
Proofs and Mathematical Reasoning University of Birmingham Author: Supervisors: Agata Stefanowicz Joe Kyle Michael Grove September 2014 c University of Birmingham 2014 Contents 1 Introduction 6 2 Mathematical language and symbols 6 2.1 Mathematics is a language . .6 2.2 Greek alphabet . .6 2.3 Symbols . .6 2.4 Words in mathematics . .7 3 What is a proof? 9 3.1 Writer versus reader . .9 3.2 Methods of proofs . .9 3.3 Implications and if and only if statements . 10 4 Direct proof 11 4.1 Description of method . 11 4.2 Hard parts? . 11 4.3 Examples . 11 4.4 Fallacious \proofs" . 15 4.5 Counterexamples . 16 5 Proof by cases 17 5.1 Method . 17 5.2 Hard parts? . 17 5.3 Examples of proof by cases . 17 6 Mathematical Induction 19 6.1 Method . 19 6.2 Versions of induction. 19 6.3 Hard parts? . 20 6.4 Examples of mathematical induction . 20 7 Contradiction 26 7.1 Method . 26 7.2 Hard parts? . 26 7.3 Examples of proof by contradiction . 26 8 Contrapositive 29 8.1 Method . 29 8.2 Hard parts? . 29 8.3 Examples . 29 9 Tips 31 9.1 What common mistakes do students make when trying to present the proofs? . 31 9.2 What are the reasons for mistakes? . 32 9.3 Advice to students for writing good proofs . 32 9.4 Friendly reminder . 32 c University of Birmingham 2014 10 Sets 34 10.1 Basics . 34 10.2 Subsets and power sets . 34 10.3 Cardinality and equality .