Lecture

Sets and Functions

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In course

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, disc ...

Description

This lecture covers the fundamental concepts of sets and functions, including union, intersection, difference, complement, symmetric difference, and set identities. It also explores set operations analogies with propositional calculus connectives, cardinality of set union, and inclusion-exclusion principles. The lecture delves into proving set identities using set builder notation, propositional logic, and membership tables. Additionally, it discusses generalized unions, intersections, and Russell's Paradox. The terminology related to functions, such as domain, codomain, range, image, and preimage, is explained. The lecture concludes with the representation of functions, injections, surjections, and bijections.

Instructor

Karl Aberer

Karl Aberer received his PhD in mathematics in 1991 from the ETH Zürich. From 1991 to 1992 he was postdoctoral fellow at the International Computer Science Institute (ICSI) at the University of California, Berkeley. In 1992, he joined the Integrated Publication and Information Systems institute (IPSI) of GMD in Germany, where he was leading the research division Open Adaptive Information Management Systems. In 2000 he joined EPFL as full professor. Since 2005 he is the director of the Swiss National Research Center for Mobile Information and Communication Systems ( NCCR-MICS, www.mics.ch ). He is member of the editorial boards of VLDB Journal, ACM Transaction on Autonomous and Adaptive Systems and World Wide Web Journal. He has been consulting for the Swiss government in research and science policy as a member of the Swiss Research and Technology Council ( SWTR ) from 2003 - 2011.

Official source

Ontological neighbourhood

Mathematics

Mathematical logic: Set theory

Logic

Classical logic: First-order logic

Related concepts (32)

Complement (set theory)

In set theory, the complement of a set A, often denoted by A∁ (or A′), is the set of elements not in A. When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A. The relative complement of A with respect to a set B, also termed the set difference of B and A, written is the set of elements in B that are not in A.

Algebra of sets

In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.

Fuzzy set

In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].

Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of elements which are in A, in B, or in both A and B.

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

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