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... First, let’s show that if [a] ⊆ [b] then [b] ⊆ [a]. Indeed, supposing that [a] ⊆ [b], let x ∈ [b], which means that Rbx (where Rbx is shorthand for hb, xi ∈ R). Since R is reflexive, a ∈ [a], and since [a] ⊆ [b], a ∈ [b]. Thus, Rba. Since R is symmetric, Rab, and since R is transitive, Rax. Therefor ...
... First, let’s show that if [a] ⊆ [b] then [b] ⊆ [a]. Indeed, supposing that [a] ⊆ [b], let x ∈ [b], which means that Rbx (where Rbx is shorthand for hb, xi ∈ R). Since R is reflexive, a ∈ [a], and since [a] ⊆ [b], a ∈ [b]. Thus, Rba. Since R is symmetric, Rab, and since R is transitive, Rax. Therefor ...
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Maximum subsets of (0,1] with no solutions to x
... is S such that x + y = z. If k is a positive integer we say that a set S of real numbers is k-sum-free if there do not exist x, y, z in S such that x + y = kz (we require that not all x, y, and z be equal to each other to avoid a meaningless problem when k = 2). Let f (n, k) denote the maximum size ...
... is S such that x + y = z. If k is a positive integer we say that a set S of real numbers is k-sum-free if there do not exist x, y, z in S such that x + y = kz (we require that not all x, y, and z be equal to each other to avoid a meaningless problem when k = 2). Let f (n, k) denote the maximum size ...
Logic and Categories As Tools For Building Theories
... products”. Once again, rather than giving a case-by-case construction of direct products in each mathematical context we encounter, we can express once and for all a general notion of product, meaningful in any category — and such that, if a product exists, it is characterised uniquely up to unique ...
... products”. Once again, rather than giving a case-by-case construction of direct products in each mathematical context we encounter, we can express once and for all a general notion of product, meaningful in any category — and such that, if a product exists, it is characterised uniquely up to unique ...
CPS130, Lecture 1: Introduction to Algorithms
... Theorem FC: The sets [And, Not], [Or, Not] and [Nand] are functionally complete where Nand(x,y) = Not(And(x,y)). proof; All we need to show is that Not, And , and Or can be computed; then we have the conclusion using Theorem G. For example, given And and Not, Or(x,y) = Not(And(Not(x), Not(y))) using ...
... Theorem FC: The sets [And, Not], [Or, Not] and [Nand] are functionally complete where Nand(x,y) = Not(And(x,y)). proof; All we need to show is that Not, And , and Or can be computed; then we have the conclusion using Theorem G. For example, given And and Not, Or(x,y) = Not(And(Not(x), Not(y))) using ...
Notes - IMSc
... P1. If `0i ≤ `i , i = 1, . . . , r, then n → (`1 , . . . , `r ) implies n → (`01 , . . . , `0r ). Clearly, if there is a monochromatic K`i , then all induced subgraphs of it of size `0i are monochromatic as well. P2. If m ≥ n and n → (`1 , . . . , `r ) then m → (`1 , . . . , `r ). This is obvious, ...
... P1. If `0i ≤ `i , i = 1, . . . , r, then n → (`1 , . . . , `r ) implies n → (`01 , . . . , `0r ). Clearly, if there is a monochromatic K`i , then all induced subgraphs of it of size `0i are monochromatic as well. P2. If m ≥ n and n → (`1 , . . . , `r ) then m → (`1 , . . . , `r ). This is obvious, ...
LOGIC AND PSYCHOTHERAPY
... 1. In 1956 Bateson and his colleagues proposed the “double-bind theory”: “...an important factor in the development of schizophrenic thought disorder is the constant subjection of an individual to a so-called double-bind situation which includes the following elements. 1) The individual has an inten ...
... 1. In 1956 Bateson and his colleagues proposed the “double-bind theory”: “...an important factor in the development of schizophrenic thought disorder is the constant subjection of an individual to a so-called double-bind situation which includes the following elements. 1) The individual has an inten ...
The Future of Post-Human Mathematical Logic
... perception, and tendered an innovative process to look at issues from a futurist's point of view. He continues on the following pages to edify his readers. Sylvan Von Burg School of Business George Washington University ...
... perception, and tendered an innovative process to look at issues from a futurist's point of view. He continues on the following pages to edify his readers. Sylvan Von Burg School of Business George Washington University ...
Adjointness in Foundations
... 1. The Formal–Conceptual Duality in Mathematics and in its Foundations2 That pursuit of exact knowledge which we call mathematics seems to involve in an essential way two dual aspects, which we may call the Formal and the Conceptual. For example, we manipulate algebraically a polynomial equation and ...
... 1. The Formal–Conceptual Duality in Mathematics and in its Foundations2 That pursuit of exact knowledge which we call mathematics seems to involve in an essential way two dual aspects, which we may call the Formal and the Conceptual. For example, we manipulate algebraically a polynomial equation and ...
Exam 1 Solutions for Spring 2014
... This is a proof by contraposition. Assume that x1 is rational. By definition of a rational number, x1 = pq for some integers p and q, with q 6= 0. We also know that x1 cannot equal 0, since there is no way to divide 1 by anything and get 0. Thus, p 6= 0. It follows that x = pq , which means that x c ...
... This is a proof by contraposition. Assume that x1 is rational. By definition of a rational number, x1 = pq for some integers p and q, with q 6= 0. We also know that x1 cannot equal 0, since there is no way to divide 1 by anything and get 0. Thus, p 6= 0. It follows that x = pq , which means that x c ...
Section 7.5: Cardinality
... countable then S is countable. This is equivalent to the statement that if S is uncountable, then X is uncountable, which is exactly the statement we are trying to prove. ...
... countable then S is countable. This is equivalent to the statement that if S is uncountable, then X is uncountable, which is exactly the statement we are trying to prove. ...
Chapter 5 Cardinality of sets
... As an aside, the vertical bars, | · |, are used throughout mathematics to denote some measure of size. For example, the absolute value of a real number measures its size in terms of how far it is from zero on the number line. According to the definition, set has cardinality n when there is a sequenc ...
... As an aside, the vertical bars, | · |, are used throughout mathematics to denote some measure of size. For example, the absolute value of a real number measures its size in terms of how far it is from zero on the number line. According to the definition, set has cardinality n when there is a sequenc ...