Sets and Venn Diagrams
... Notation: For A a set, n(A) means “the number of elements in the set A”. Examples: If A = {7, 24, 2.5}, then n(A) = 3. For B = { 0 }, n(B) = 1, while n(∅) = 0. In fact, for any set, C, if n(C) = 0, then C = ∅. ...
... Notation: For A a set, n(A) means “the number of elements in the set A”. Examples: If A = {7, 24, 2.5}, then n(A) = 3. For B = { 0 }, n(B) = 1, while n(∅) = 0. In fact, for any set, C, if n(C) = 0, then C = ∅. ...
Set Concepts
... Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call ...
... Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call ...
22.1 Representability of Functions in a Formal Theory
... 1930’s. A variety of formal models has been developed, such as the λ-calculus (functional programming), Turing machines (imperative programming), recursive functions, and a huge number of programming languages. All these models have turned out to be equivalent. None is stronger than the others, so w ...
... 1930’s. A variety of formal models has been developed, such as the λ-calculus (functional programming), Turing machines (imperative programming), recursive functions, and a huge number of programming languages. All these models have turned out to be equivalent. None is stronger than the others, so w ...
BASIC SET THEORY
... it. Nothing can be "half in" , "in twice", etc. {A,B,C} and {A,A,B,C} are the same set. If set A and set B have the same elements, then by definition they are two names for the SAME SET. If either has an element the other doesn't, they are DIFFERENT SETS. This is called the "extensive property" of s ...
... it. Nothing can be "half in" , "in twice", etc. {A,B,C} and {A,A,B,C} are the same set. If set A and set B have the same elements, then by definition they are two names for the SAME SET. If either has an element the other doesn't, they are DIFFERENT SETS. This is called the "extensive property" of s ...
docx
... contains all pairs from both sets. o This idea shows up in a lot places, including databases (all pair of two lists…) o That last line is there to make it clear that I can take the Cartesian product of many sets. ...
... contains all pairs from both sets. o This idea shows up in a lot places, including databases (all pair of two lists…) o That last line is there to make it clear that I can take the Cartesian product of many sets. ...
Slide 1
... • The checking algorithm A would then verify that the tour really does visit all of the cities and really does have total length K. without seeking all possible K solutions through each of the vertices. Polynomial. • The TSP, therefore, also belongs to NP. • How could a problem fail to belong to NP? ...
... • The checking algorithm A would then verify that the tour really does visit all of the cities and really does have total length K. without seeking all possible K solutions through each of the vertices. Polynomial. • The TSP, therefore, also belongs to NP. • How could a problem fail to belong to NP? ...
Proposition: The following properties hold A ∩ B ⊆ A, A ∩ B ⊆ B, A
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
... Definition: If A is a finite set we write |A| for the number of elements in A and call |A| the cardinality of A. Proposition: If A and B are finite sets then |A ∪ B| = |A| + |B| − |A ∩ B| Proof: From a Venn diagram we see |A ∪ B| = |A∩ ∼B| + |A ∩ B| + | ∼A ∩ B| The first two terms give |A| while the ...
SETS - Hatboro
... To specify a set, you must identify its elements. Frequently you can specify a set by listing or making a roster of its elements. For example, the set of vowels of the English alphabet can be represented in roster form by {a,e,i,o,u}. A second way of specifying a set is by giving a rule or condition ...
... To specify a set, you must identify its elements. Frequently you can specify a set by listing or making a roster of its elements. For example, the set of vowels of the English alphabet can be represented in roster form by {a,e,i,o,u}. A second way of specifying a set is by giving a rule or condition ...
2.1 Symbols and Terminology
... • The intersection of sets A and B, written A ∩ B is the set of elements common to both A and B. In symbols we write A ∩ B = {x | x ∈ A and x ∈ B}. • Disjoint sets are sets that have no elements in common. Specifically A ∩ B = ∅. • The union of sets A and B, written A ∪ B is the set of all elements b ...
... • The intersection of sets A and B, written A ∩ B is the set of elements common to both A and B. In symbols we write A ∩ B = {x | x ∈ A and x ∈ B}. • Disjoint sets are sets that have no elements in common. Specifically A ∩ B = ∅. • The union of sets A and B, written A ∪ B is the set of all elements b ...
Readings for Lecture/Lab 1 – Sets and Whole Numbers How are the
... How many other distinct one-to-one correspondences could be made where a, b, c are kept in the same order? What are they? That is, how many different one-to-one correspondences could be made? Important Note. Equal sets are equivalent, but equivalent sets may not be equal. This was illustrated in the ...
... How many other distinct one-to-one correspondences could be made where a, b, c are kept in the same order? What are they? That is, how many different one-to-one correspondences could be made? Important Note. Equal sets are equivalent, but equivalent sets may not be equal. This was illustrated in the ...
Lec2Logic
... -For all x and for all y if x is positive and y is negative then their product must be negative. -The product of a positive and a negative real number is negative. Translate this sentence into a logical expressions. “If a person is female and is a parent, then she is someone’s mother.” F(x) is “x is ...
... -For all x and for all y if x is positive and y is negative then their product must be negative. -The product of a positive and a negative real number is negative. Translate this sentence into a logical expressions. “If a person is female and is a parent, then she is someone’s mother.” F(x) is “x is ...
PDF
... (functional programming), Turing machines (imperative programming), recursive functions, and a huge number of programming languages. All these models have turned out to be equivalent. None is stronger than the others, so we can be quite sure that they do represent the class of all computable functio ...
... (functional programming), Turing machines (imperative programming), recursive functions, and a huge number of programming languages. All these models have turned out to be equivalent. None is stronger than the others, so we can be quite sure that they do represent the class of all computable functio ...
Number systems. - Elad Aigner
... Let n∗ denote this element and let m∗ ∈ Z+ be a positive integer such that {m∗ , n∗ } ∈ S. Put another way {m∗ , n∗ } is a pair in S whose n-part is least amongst all pairs in S. To get a contradiction we show that there exists a pair {p, q} ∈ S such that q < n∗ . How can we come up with the numbers ...
... Let n∗ denote this element and let m∗ ∈ Z+ be a positive integer such that {m∗ , n∗ } ∈ S. Put another way {m∗ , n∗ } is a pair in S whose n-part is least amongst all pairs in S. To get a contradiction we show that there exists a pair {p, q} ∈ S such that q < n∗ . How can we come up with the numbers ...
PPT
... Let h(x) be computed by a program with number p. Then p TOT, which means that p = g(i) for some i. Then h(i) = (i, g(i)) + 1 by definition of h ...
... Let h(x) be computed by a program with number p. Then p TOT, which means that p = g(i) for some i. Then h(i) = (i, g(i)) + 1 by definition of h ...
slides - Department of Computer Science
... So we get: PA, except that axioms assert only the existence of finite sets definable with formulas (formulas with no string-quantifiers and with bounded number-quantifiers.) Such formulas correspond to a (weak) complexity class: constant-depth Boolean circuits of polynomial-size (aka AC0). Denote th ...
... So we get: PA, except that axioms assert only the existence of finite sets definable with formulas (formulas with no string-quantifiers and with bounded number-quantifiers.) Such formulas correspond to a (weak) complexity class: constant-depth Boolean circuits of polynomial-size (aka AC0). Denote th ...
a note on the recursive unsolvability of primitive recursive arithmetic
... th(fe) =sub(i, i) is valid and hence provable. But if this formula and (2) are provable then by modus ponens and substitution (Ez). z^k &f(z) = sub(i, i). Thus sub(i, i) is one of the first k nontheorems, contrary to hypothesis. Suppose (2) is not a theorem. By hypothesis,/enumerates all nontheorems ...
... th(fe) =sub(i, i) is valid and hence provable. But if this formula and (2) are provable then by modus ponens and substitution (Ez). z^k &f(z) = sub(i, i). Thus sub(i, i) is one of the first k nontheorems, contrary to hypothesis. Suppose (2) is not a theorem. By hypothesis,/enumerates all nontheorems ...
2.1 Notes
... (a) The set is finite since the letters used are C, D, I, L, M, V, and X. (b) The set is infinite since it consists of an unlimited number of elements. (c) The set is finite since there is a specific number of people in your immediate family. (d) The set is infinite because an unlimited number of so ...
... (a) The set is finite since the letters used are C, D, I, L, M, V, and X. (b) The set is infinite since it consists of an unlimited number of elements. (c) The set is finite since there is a specific number of people in your immediate family. (d) The set is infinite because an unlimited number of so ...
06. Naive Set Theory
... Russell’s Paradox is a paradox of the One and the Many: It looks like R can’t be thought of as a “one”. SO: Sets were introduced initially (in part) to address paradoxes of the Infinitely Big. But now it seems we’ve just replaced them with paradoxes of the One and the Many! ...
... Russell’s Paradox is a paradox of the One and the Many: It looks like R can’t be thought of as a “one”. SO: Sets were introduced initially (in part) to address paradoxes of the Infinitely Big. But now it seems we’ve just replaced them with paradoxes of the One and the Many! ...