A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right

1.A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. A set is a well defined collection of objects. Georg Cantor, the founder of set theory, gave the following definition of a set :A set is a gathering together into a whole of definite, distinct objects of our perception and of our thought – which are called elements of the set. The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets A and B are equal if and only if they have precisely the same elements. The most basic properties are that a set "has" elements, and that two sets are equal if and only if every element of one is an element of the other. Writing A = {1, 2, 3, 4 } means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example {1, 2}, are subsets of A. Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}. The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colors red, green and blue. The relation "is an element of", also called set membership, is denoted by ∈. Writing means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", " is in A" and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, however some authors use them to mean instead "x is a subset of A". The number of elements in a particular set is a property known as cardinality, informally this is the size of a set. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3. An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of natural numbers, N = { 1, 2, 3, 4, ... }. In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.

A is a proper subset of B and conversely B is a proper superset of A

In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets.

If every member of set A is also a member of set B, then A is said to be a subset of B, written A ⊆ B (A is contained in B). Equivalently, we can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment. If A is a subset of, but not equal to, B, then A is called a proper subset of B, written A ⊊ B (A is a proper subset of B) or B ⊋ A (B is a proper superset of A).Example:

· The set of all men is a proper subset of the set of all people.

· {1, 3} ⊊ {1, 2, 3, 4}.

· {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set and every set is a subset of itself:

· ∅ ⊆ A.

· A ⊆ A.

An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:

· A = B if and only if A ⊆ B and B ⊆ A.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets.

 

 

2. In mathematics, an ordered pair (a, b) is a pair of mathematical objects. In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. Alternatively, the objects are called the first and second coordinates, or the left and right projections of the ordered pair. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. This attribute is opposite of unordered pair's attribute: the unordered pair {a, b} equals the unordered pair {b, a}.

Ordered pairs are also called 2-tuples, 2-dimensional vectors, or sequences of length 2. The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another. Cartesian products and binary relations are defined in terms of ordered pairs.

In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an (ordered) -tuple is a sequence (or ordered list) of elements, where is a positive integer. One 0-tuple, an empty sequence, also exists. An -tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple. Sometimes other delimiters are used, such as square brackets "" or angle brackets "". Braces "" are almost never used for tuples, as they are the standard notation for sets.

Tuples are often used to describe other mathematical objects, such as vectors. In algebra, a ring is commonly defined as a 3-tuple , where is some set, and "", and "" are functions mapping the Cartesian product to with specific properties. In computer science, tuples are directly implemented as product types in most functional programming languages.

Tuples as nested ordered pairs

1. The 0-tuple (i.e. the empty tuple) is represented by the empty set . 2. An -tuple, with , can be defined as an ordered pair of its first entry and…

Examples

Non-injective and surjective

· 4. In mathematics, the power set (or powerset) of any set S, written , P(S),… Any subset of is called a family of sets over S.

Definable relations

In other words, R is definable if and only if there is a formula φ such…

Edit]Definability with parameters

A relation R is said to be definable with parameters (or -definable) if there is a formula φ with parameters from such that R is definable using φ. Every element of a structure is definable using the element itself as a parameter.

 

König's Lemma

König's lemma states that a finitely branching tree is infinite iff it has an infinite path. This lemma is used iKönig's lemma or König's infinity lemma is a… If G is a connected graph with infinitely many vertices such that every vertex has finite degree (that is, each vertex…

Proof

For the proof, assume that the graph consists of infinitely many vertices and is connected.

Start with any vertex v₁. Every one of the infinitely many vertices of G can be reached from v₁ with a simple path, and each such path must start with one of the finitely many vertices adjacent to v₁. There must be one of those adjacent vertices through which infinitely many vertices can be reached without going through v₁. If there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite. We may thus pick one of these vertices and call it v₂.

Now infinitely many vertices of G can be reached from v₂ with a simple path which doesn't use the vertex v₁. Each such path must start with one of the finitely many vertices adjacent to v₂. So an argument similar to the one above shows that there must be one of those adjacent vertices through which infinitely many vertices can be reached; pick one and call it v₃.

Continuing in this fashion, an infinite simple path can be constructed by mathematical induction. At each step, the induction hypothesis states that there are infinitely many nodes reachable by a simple path from a particular node that does not go through one of a finite set of vertices. The induction argument is that one of the vertices adjacent to satisfies the induction hypothesis, even when is added to the finite set.

The result of this induction argument is that for all n it is possible to choose a vertex v_n as the construction describes. The set of vertices chosen in the construction is then a chain in the graph, because each one was chosen to be adjacent to the previous one, and the construction guarantees that the same vertex is never chosen twice.

This proof may not be considered constructive, because at each step it uses a proof by contradiction to establish that there exists an adjacent vertex from which infinitely many other vertices can be reached. Facts about the computational aspects of the lemma suggest that no proof can be given that would be considered constructive by the main schools of constructive mathematics.

 

 

7-8.

9. In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance. A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of associated mathematical objects is called the structure and the set is called the underlying set.

 

 

Characteristic properties of tuples

if and only if Thus a tuple has properties that distinguish it from a set. 1. A tuple may contain multiple instances of the same element, so tuple ; but set = .

Associativity

The associative property is closely related to the commutative property. The associative property of an expression containing two or more… Most commutative operations encountered in practice are also associative.…

Examples Commutative operations in mathematics

· The addition of real numbers is commutative, since For example 4 + 5 = 5 + 4, since both expressions equal 9.

Edit]Examples

Edit]Simple example

Let be a set with an equivalence relation . For this relation are equivalence classes:

Set of all equivalence classes for this relation is .

Equivalence class

The set of all a and b for which a ~ b holds make up an equivalence class of X by ~. Let denote the equivalence class to which a belongs. Then all…    

Definitions

In classical mathematics, one may see these versions of asymmetry and connectedness: x≮y or y≮x; x<y or y<x or x=y.   15. In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x <…