Lecture

Cartesian Product

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Description

This lecture introduces the concept of the Cartesian product, which is a set that contains all possible ordered pairs of elements from two sets. The instructor explains the importance of the order of elements in a pair and provides examples to illustrate the concept. The lecture covers the definition of the Cartesian product for multiple sets and discusses relations as subsets of the Cartesian product. Various examples and general definitions are presented to deepen the understanding of this fundamental concept in set theory.

In MOOCs (2)

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

Analysis I (part 1): Prelude, basic concepts, real numbers

Can you calculate the value of sin(π/4)? Does the identity ln(xy)=ln(x)+ln(y) sound familiar? In the prelude we recall and expand on a few such notions, in particular the trigonometric functions sin, cos and tan and their properties, the concept of reciprocal functions such as exp and ln, some calculation rules concerning powers, logarithms and roots, sets and functions. Next we discuss number systems: based on the "intuitive" notion of the natural integers N={0,1,2,3,...}, we rigorously define the rational

Official source

Quiz

Ontological neighbourhood

Mathematics

Mathematical logic: Set theory

Related concepts (31)

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.

Naive set theory

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.

Cartesian product

In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).

Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects. 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. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors.

Zermelo set theory

Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.

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