Lecture 22 - Duke Computer Science
... The rationals are dense: between any two there is a third. You can’t list them one by one without leaving out an infinite number of them ...
... The rationals are dense: between any two there is a third. You can’t list them one by one without leaving out an infinite number of them ...
The Ring of Integers
... Elementary number theory is largely about the ring of integers, denoted by the symbol Z. The integers are an example of an algebraic structure called an integral domain. This means that Z satisfies the following axioms: (a) Z has operations + (addition) and · (multiplication). It is closed under the ...
... Elementary number theory is largely about the ring of integers, denoted by the symbol Z. The integers are an example of an algebraic structure called an integral domain. This means that Z satisfies the following axioms: (a) Z has operations + (addition) and · (multiplication). It is closed under the ...
Section 2.2 Subsets
... empty set) or n(A) is a natural number. • Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number א0 read “aleph-null”. • Equal sets: Set A is equal to set B if set A and set B contain exactly the same elements, regard ...
... empty set) or n(A) is a natural number. • Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number א0 read “aleph-null”. • Equal sets: Set A is equal to set B if set A and set B contain exactly the same elements, regard ...
On fuzzy semi-preopen sets and fuzzy semi
... In the paper by (X, ) or simply by X we mean an I-topological space. intA, clA and Ac denote the interior, closure and complement of a subset A in X, respectively. A subset A in X is called I-preopen if and only if A int(clA), and I-preclosed if and only if A cl(intA) [3,6]. A subset A in X is calle ...
... In the paper by (X, ) or simply by X we mean an I-topological space. intA, clA and Ac denote the interior, closure and complement of a subset A in X, respectively. A subset A in X is called I-preopen if and only if A int(clA), and I-preclosed if and only if A cl(intA) [3,6]. A subset A in X is calle ...
The Number of Topologies on a Finite Set
... Sharp [1] proved this result using graph theory. Stephen’s proof [4]used topological facts. Here is another one, which is direct and allows us to compute T (n, 3 · 2n−2 ). Proof. Since we are looking for a non-discrete topology τ having the maximum of open sets, it must not contain all the singleton ...
... Sharp [1] proved this result using graph theory. Stephen’s proof [4]used topological facts. Here is another one, which is direct and allows us to compute T (n, 3 · 2n−2 ). Proof. Since we are looking for a non-discrete topology τ having the maximum of open sets, it must not contain all the singleton ...
22.1 Representability of Functions in a Formal Theory
... p(x)=y iff x=0 and y=0 or if x=y+1. Based on this analysis we define the predicate R p as Rp (x,y) ≡ (x=0 ∧ y=0) ∨ y+1=x. What remains to show is that this representation is in fact correct, that is that p(x)=y implies the validity of Rp (x,y) in Peano Arithmetic and that p(x)6=y implies the validit ...
... p(x)=y iff x=0 and y=0 or if x=y+1. Based on this analysis we define the predicate R p as Rp (x,y) ≡ (x=0 ∧ y=0) ∨ y+1=x. What remains to show is that this representation is in fact correct, that is that p(x)=y implies the validity of Rp (x,y) in Peano Arithmetic and that p(x)6=y implies the validit ...
A(x)
... Could the formula F have a finite model? U = {a1, a2, a3, ... ? } To a1 there must exist an element a2, so that P(a1, a2), a2 a1 To a2 there must exist an element a3 such that P(a2, a3), a3 a2, and a3 a1 otherwise P(a1, a2) P(a2, a1), so P(a1, a1). To a3 there must exist an element a4 such t ...
... Could the formula F have a finite model? U = {a1, a2, a3, ... ? } To a1 there must exist an element a2, so that P(a1, a2), a2 a1 To a2 there must exist an element a3 such that P(a2, a3), a3 a2, and a3 a1 otherwise P(a1, a2) P(a2, a1), so P(a1, a1). To a3 there must exist an element a4 such t ...
4. Techniques of Proof: II
... Letting f (k) = sk for 1 k n, we obtain the more familiar notation S = {s1, s2, …, sn}. The same kind of counting process is possible for a denumerable set, and this is why both kinds of sets are called countable. For example, if T is denumerable, then there exists a bijection g : T, and we ma ...
... Letting f (k) = sk for 1 k n, we obtain the more familiar notation S = {s1, s2, …, sn}. The same kind of counting process is possible for a denumerable set, and this is why both kinds of sets are called countable. For example, if T is denumerable, then there exists a bijection g : T, and we ma ...
Proof Theory - Andrew.cmu.edu
... parts of ordinary mathematics, but weak enough, on the other hand, to be amenable to proof-theoretic analysis. He then suggested “calibrating” various mathematical theorems in terms of their axiomatic strength. Whereas in ordinary (meta)mathematics, one proves theorems from axioms, Friedman noticed ...
... parts of ordinary mathematics, but weak enough, on the other hand, to be amenable to proof-theoretic analysis. He then suggested “calibrating” various mathematical theorems in terms of their axiomatic strength. Whereas in ordinary (meta)mathematics, one proves theorems from axioms, Friedman noticed ...