On the determination of sets by the sets of sums of a certain order
... that the numbers Fs(n) remain the same even if we restrict ourselves to sets of positive integers. Thus the results in [5] remain valid in our case. These included a necessary condition for Fs(n) > 1 and the proof that F2(n) > 1 (and hence Fn_2(n) > 1) if and only if n is a power of 2. Also Fs(2s) > ...
... that the numbers Fs(n) remain the same even if we restrict ourselves to sets of positive integers. Thus the results in [5] remain valid in our case. These included a necessary condition for Fs(n) > 1 and the proof that F2(n) > 1 (and hence Fn_2(n) > 1) if and only if n is a power of 2. Also Fs(2s) > ...
Chapter 1: Sets, Operations and Algebraic Language
... Note that the dot on the number line where the number 1 is located is open in the first example and closed in the second example. The open dot indicates that the number 1 is not an element of the set of all numbers greater than 1. The closed dot on the second number line indicates that 1 is an eleme ...
... Note that the dot on the number line where the number 1 is located is open in the first example and closed in the second example. The open dot indicates that the number 1 is not an element of the set of all numbers greater than 1. The closed dot on the second number line indicates that 1 is an eleme ...
Notes on Infinite Sets
... that we can show is 1-1 and onto. Since each integer m falls uniquely between two successive powers of 2, given m, there is only one corresponding power of 2 in this definition that “brackets” m. And each m clearly has a unique n-bit binary expansion, so f is 1-1. To see that f is onto, pick any str ...
... that we can show is 1-1 and onto. Since each integer m falls uniquely between two successive powers of 2, given m, there is only one corresponding power of 2 in this definition that “brackets” m. And each m clearly has a unique n-bit binary expansion, so f is 1-1. To see that f is onto, pick any str ...
Chapter 4 Set Theory
... “A set is a Many that allows itself to be thought of as a One.” (Georg Cantor) In the previous chapters, we have often encountered ”sets”, for example, prime numbers form a set, domains in predicate logic form sets as well. Defining a set formally is a pretty delicate matter, for now, we will be hap ...
... “A set is a Many that allows itself to be thought of as a One.” (Georg Cantor) In the previous chapters, we have often encountered ”sets”, for example, prime numbers form a set, domains in predicate logic form sets as well. Defining a set formally is a pretty delicate matter, for now, we will be hap ...
ch42 - Kent State University
... • It shows that some languages are not decidable or even Turing-recognizable, for the reason that there are uncountably many languages yet only countably many Turing machines. • Since each Turing machine can recognize a single language and there are more languages than Turing machines, some language ...
... • It shows that some languages are not decidable or even Turing-recognizable, for the reason that there are uncountably many languages yet only countably many Turing machines. • Since each Turing machine can recognize a single language and there are more languages than Turing machines, some language ...
accept accept accept accept
... • It shows that some languages are not decidable or even Turing-recognizable, for the reason that there are uncountably many languages yet only countably many Turing machines. • Since each Turing machine can recognize a single language and there are more languages than Turing machines, some language ...
... • It shows that some languages are not decidable or even Turing-recognizable, for the reason that there are uncountably many languages yet only countably many Turing machines. • Since each Turing machine can recognize a single language and there are more languages than Turing machines, some language ...
HW 12
... 4. The difference between two sets A and B is the set of all objects that belong to set A but not to B. This is written as A \ B a. Provide a definitional axiom for A \ B (use a 2-place function symbol diff(x,y)) b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A ...
... 4. The difference between two sets A and B is the set of all objects that belong to set A but not to B. This is written as A \ B a. Provide a definitional axiom for A \ B (use a 2-place function symbol diff(x,y)) b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A ...
Section 1.1
... (b) Woolly mammoths have been extinct for almost 8,000 years, so this set is most definitely empty. (c) Be careful! Each instance of { } and represents the empty set, but {} is a set with one element: . (d) This set is empty because there are no natural numbers between 1 and 2. ...
... (b) Woolly mammoths have been extinct for almost 8,000 years, so this set is most definitely empty. (c) Be careful! Each instance of { } and represents the empty set, but {} is a set with one element: . (d) This set is empty because there are no natural numbers between 1 and 2. ...
Exercises about Sets
... Write the sets with the roster method, set builder notation, and interval notation whenever possible. How many elements are in each set? Sketch the graph of each set on a Real number line. Use Venn diagrams to show the relationships between the sets involved in each question. a) N (2,5) b) N (-2 ...
... Write the sets with the roster method, set builder notation, and interval notation whenever possible. How many elements are in each set? Sketch the graph of each set on a Real number line. Use Venn diagrams to show the relationships between the sets involved in each question. a) N (2,5) b) N (-2 ...
Mathematical Ideas - Millersville University of Pennsylvania
... A one-to-one correspondence between two sets is a pairing where each element of one set is paired with exactly one element of the second set and each element of the second set is paired with exactly one element of the first set. ...
... A one-to-one correspondence between two sets is a pairing where each element of one set is paired with exactly one element of the second set and each element of the second set is paired with exactly one element of the first set. ...
Functional Programming and the Lambda Calculus
... Pure lambda calculus has no built-in functions; we’ll be impure. To evaluate (+ (∗ 5 6) (∗ 8 3)), we can’t start with + because it only operates on numbers. There are two reducible expressions: (∗ 5 6) and (∗ 8 3). We can reduce either one first. For example: ...
... Pure lambda calculus has no built-in functions; we’ll be impure. To evaluate (+ (∗ 5 6) (∗ 8 3)), we can’t start with + because it only operates on numbers. There are two reducible expressions: (∗ 5 6) and (∗ 8 3). We can reduce either one first. For example: ...
PPT - Carnegie Mellon School of Computer Science
... A Natural Intuition Intuitively, what does it mean to find a bijection between a set A and ? It means to list the elements of A in some order so that if you read down the list, every element will get read. ...
... A Natural Intuition Intuitively, what does it mean to find a bijection between a set A and ? It means to list the elements of A in some order so that if you read down the list, every element will get read. ...
4 Sets and Operations on Sets
... (a) Exhibit all the one-to-one correspondence between the two sets. (b) How many such one-to-one correspondence are there? Solution. (a) Figure 4.5 shows all the one-to-one correspondence between the two sets. (b) There are six one-to-one correspondence. ...
... (a) Exhibit all the one-to-one correspondence between the two sets. (b) How many such one-to-one correspondence are there? Solution. (a) Figure 4.5 shows all the one-to-one correspondence between the two sets. (b) There are six one-to-one correspondence. ...
Infinity + Infinity
... Firstly, addition is an operation only for numbers, which ∞, in the traditional sense is not. We must look to the arithmetic of cardinal numbers to make sense of ∞+∞. Further, when discussing infinity, it is imperative to know what ”infinity” we are talking about. For any grade school student, the i ...
... Firstly, addition is an operation only for numbers, which ∞, in the traditional sense is not. We must look to the arithmetic of cardinal numbers to make sense of ∞+∞. Further, when discussing infinity, it is imperative to know what ”infinity” we are talking about. For any grade school student, the i ...
Document
... If A and B are subsets of S, then the cartesian product (cross product) of A and B denoted symbolically by A B is defined by A B = {(x,y) | x A and y B } The Cartesian product of 2 sets is the set of all combinations of ordered pairs that can be produced from the elements of both sets. Examp ...
... If A and B are subsets of S, then the cartesian product (cross product) of A and B denoted symbolically by A B is defined by A B = {(x,y) | x A and y B } The Cartesian product of 2 sets is the set of all combinations of ordered pairs that can be produced from the elements of both sets. Examp ...
sets and elements
... A set could have as many entries as you would like. It could have one entry, 10 entries, 15 entries, infinite number of entries, or even have no entries at all! For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of e ...
... A set could have as many entries as you would like. It could have one entry, 10 entries, 15 entries, infinite number of entries, or even have no entries at all! For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of e ...
Section 2.2
... Quite often, in order to determine the cardinality of a set, it is easiest to determine the cardinality of another set with which we know it is in bijective correpsondence. Example 2.2.8. How many integers between 1 and 1000 are perfect squares? Solution: The list of perfect squares in our range beg ...
... Quite often, in order to determine the cardinality of a set, it is easiest to determine the cardinality of another set with which we know it is in bijective correpsondence. Example 2.2.8. How many integers between 1 and 1000 are perfect squares? Solution: The list of perfect squares in our range beg ...
Document
... If A and B are subsets of S, then the cartesian product (cross product) of A and B denoted symbolically by A B is defined by A B = {(x,y) | x A and y B } The Cartesian product of 2 sets is the set of all combinations of ordered pairs that can be produced from the elements of both sets. Examp ...
... If A and B are subsets of S, then the cartesian product (cross product) of A and B denoted symbolically by A B is defined by A B = {(x,y) | x A and y B } The Cartesian product of 2 sets is the set of all combinations of ordered pairs that can be produced from the elements of both sets. Examp ...
Recursion, Scope, Function Templates
... coworkers (clones) who work and act like you do Ask clone to solve a simpler but similar problem Use clone’s result to put together your answer Looks like calling a function in itself, but should be done very ...
... coworkers (clones) who work and act like you do Ask clone to solve a simpler but similar problem Use clone’s result to put together your answer Looks like calling a function in itself, but should be done very ...
full text (.pdf)
... where C and D are disjoint finite subsets of N. We take T to be the set of logical consequences of T0 and the formulas (2). Every Herbrand model of T0 extends to a model of T , because new elements outside the Herbrand domain can be freely added as needed to satisfy the existential formulas (2). To s ...
... where C and D are disjoint finite subsets of N. We take T to be the set of logical consequences of T0 and the formulas (2). Every Herbrand model of T0 extends to a model of T , because new elements outside the Herbrand domain can be freely added as needed to satisfy the existential formulas (2). To s ...
mathematical logic: constructive and non
... However, if we agree here that a c proof ' of a sentence should be a finite linguistic construction, recognizable as being made in accordance with preassigned rules and whose existence assures the 'truth' of the sentence in the appropriate sense, we already have (II ), since the verification of (2) ...
... However, if we agree here that a c proof ' of a sentence should be a finite linguistic construction, recognizable as being made in accordance with preassigned rules and whose existence assures the 'truth' of the sentence in the appropriate sense, we already have (II ), since the verification of (2) ...
Sets, Functions, Relations - Department of Mathematics
... working with strings we will use a similar notation with a different meaning—be careful not to confuse it. 2When ...
... working with strings we will use a similar notation with a different meaning—be careful not to confuse it. 2When ...
chap1sec7 - University of Virginia, Department of Computer
... We have a table of set identities just as we had a table of logical equivalences. This is table 1 on page 89 of the text. Take a look at this table and compare it to our table for logical equivalences. You will find them to be strikingly similar. This is not a coincidence. Every set operation that w ...
... We have a table of set identities just as we had a table of logical equivalences. This is table 1 on page 89 of the text. Take a look at this table and compare it to our table for logical equivalences. You will find them to be strikingly similar. This is not a coincidence. Every set operation that w ...
Mathematical Proofs - Kutztown University
... Example: The set of all positive even integers less than 41 can be described by X={2, 4, …, 40} The set of all positive odd integers can be described by Y={1, 3, 5, …} ...
... Example: The set of all positive even integers less than 41 can be described by X={2, 4, …, 40} The set of all positive odd integers can be described by Y={1, 3, 5, …} ...
Lecture 5 MATH1904 • Disjoint union If the sets A and B have no
... If one thing can be selected in a ways and another thing can be selected in b ways, then the number of different ways of selecting the first and the second thing is ab. This principle actually goes beyond the formula for |A × B| because the set from which the second choice is made could depend on th ...
... If one thing can be selected in a ways and another thing can be selected in b ways, then the number of different ways of selecting the first and the second thing is ab. This principle actually goes beyond the formula for |A × B| because the set from which the second choice is made could depend on th ...