Mathematical Proofs - Kutztown University
... A set is a collection of objects. The objects that make up the set are called its elements. It’s customary to use capital (uppercase) letters (such as A, B, C, S, X, Y) to describe sets and lowercase letters (for example, a, b, c, s, x, y) to represent elements of sets. If a is an element of the set ...
... A set is a collection of objects. The objects that make up the set are called its elements. It’s customary to use capital (uppercase) letters (such as A, B, C, S, X, Y) to describe sets and lowercase letters (for example, a, b, c, s, x, y) to represent elements of sets. If a is an element of the set ...
Chapter 4 Set Theory
... justify than what we had intuitively before: both sets are equal because whenever a number belongs to one, it belongs to the other. Definition 28. The cardinality of a set S is the number of distinct elements of S. If |S| is finite, the set is said to be finite. It is said to be infinite otherwise. ...
... justify than what we had intuitively before: both sets are equal because whenever a number belongs to one, it belongs to the other. Definition 28. The cardinality of a set S is the number of distinct elements of S. If |S| is finite, the set is said to be finite. It is said to be infinite otherwise. ...
Lesson 1 – Types of Sets and Set Notation
... (you can use any variable) 2) Put your smallest number in the first blank and your largest number in the last blank 3) This is read as: Blank is less than or equal to x which is less than or equal to blank. This means that whatever you choose for x has to between the two blanks. Representing Using a ...
... (you can use any variable) 2) Put your smallest number in the first blank and your largest number in the last blank 3) This is read as: Blank is less than or equal to x which is less than or equal to blank. This means that whatever you choose for x has to between the two blanks. Representing Using a ...
Lecture notes 2 -- Sets
... We can give the above set a name, which will be helpful when we want to reference it later: S = {1, 2, 3, 4, 5}. ...
... We can give the above set a name, which will be helpful when we want to reference it later: S = {1, 2, 3, 4, 5}. ...
PHI 312
... A finite set of points in the plane, no point in the set lying on the line connecting any two other points in the set, is said to be convex if no point lies in the inside of any triangle whose vertices are also points in the set. It is a known result of geometry that any set of at least 5 points in ...
... A finite set of points in the plane, no point in the set lying on the line connecting any two other points in the set, is said to be convex if no point lies in the inside of any triangle whose vertices are also points in the set. It is a known result of geometry that any set of at least 5 points in ...
Introduction to Sets and Functions
... irrational numbers are all infinite. Not all infinite sets are considered to be the same “size.” The set of real numbers is considered to be a much larger set than the set of integers. In fact, this set is so large that we cannot possibly list all its elements in any organized manner the way the int ...
... irrational numbers are all infinite. Not all infinite sets are considered to be the same “size.” The set of real numbers is considered to be a much larger set than the set of integers. In fact, this set is so large that we cannot possibly list all its elements in any organized manner the way the int ...
when you hear the word “infinity”? Write down your thoughts and
... The dots indicate that we continue the sequence. Dots with no number after them means the sequence goes on forever. Examples: ...
... The dots indicate that we continue the sequence. Dots with no number after them means the sequence goes on forever. Examples: ...
A NATURAL REPRESENTATION OF BOUNDED LATTICES There
... / Ax then 1.1 yields q ∈ D(L) such that p−1 (0) ⊆ q −1 (0) and q(x) = 1, hence ...
... / Ax then 1.1 yields q ∈ D(L) such that p−1 (0) ⊆ q −1 (0) and q(x) = 1, hence ...
MATH 2420 Discrete Mathematics
... A ⊆ B ⇔ ∀ x, if x ∈ A then x ∈ B. The definition of subset is rigid and inflexible. If any element in A does not appear in B then A cannot be a subset of B. That is: A 6⊆ B ⇔ ∃ x such that x ∈ A and x 6∈ B. Looking at the special sets above we have ...
... A ⊆ B ⇔ ∀ x, if x ∈ A then x ∈ B. The definition of subset is rigid and inflexible. If any element in A does not appear in B then A cannot be a subset of B. That is: A 6⊆ B ⇔ ∃ x such that x ∈ A and x 6∈ B. Looking at the special sets above we have ...
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 = ∅. ...
Application of Optimal Sampling Lattices on CT Image
... points of a regular hexagonal structure is just about 87% of the number of lattice points of a square structure. Hence the CT reconstruction time will be reduced if the reconstruction is done on the regular hexagonal domain instead of the square domain. From the figure shown in the last slide, we ...
... points of a regular hexagonal structure is just about 87% of the number of lattice points of a square structure. Hence the CT reconstruction time will be reduced if the reconstruction is done on the regular hexagonal domain instead of the square domain. From the figure shown in the last slide, we ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.