"The first thing G.K. Cantor, who developed set theory, noticed was that there are different sizes of infinity."
"Cantor had made a very important observation: there are different ways to generate larger infinities."
"So we start with the open set, we look at all the what all the things in the open set, what's best."
"There are infinite sizes of infinity, and Cantor demonstrated this."
"If there's a stable cycle to be found, that one seed value is definitely going to find it."
"If a point eventually falls into some stable predictable pattern, we say that it's part of the Fatou set of our iterated function."
"Russell's paradox shows that there is a logical problem with trying to categorize anything into coherent sets."
"Set containment defines a partial order."
"The set of all possible infinite binary sequences is uncountable."
"The rules of set theory are not made-up rules; they're real, non-made-up, objective rules that already exist and they govern perhaps one of the most fundamental practices of human existence, and that is predication."
"The axiom of choice doesn't actually give us a way to find this function or what this function looks like, it just declares that for any set of nonempty sets X such a function must exist."
"It's the set containing 'a' and 'b' where the order matters."
"...mathematics is not purely reducible to logic, you have to supplement it with extra assumptions perhaps assumptions from set theory."
"The axiom of choice is fundamentally different from the other axioms of set theory. All the other axioms of set theory have a set existence claim for sets that have a property defined by the features expressed in the axiom itself and they're unique."
"A group is a set, a set of things."
"How many distinct integers from the set 1 two three and so on through 20 must be chosen to guarantee that two of them have a sum of 25?"
"If a set has an infinite number of elements like the set of prime numbers, then it's perfectly fine to write the infinity symbol as the cardinality of the set."
"If two sets are equal, they both contain the same elements."
"A really nice and intuitive property of subsets is that if A is a subset of B and B is a subset of C, then A is a subset of C."
"The empty set is a set which contains no elements."
"The universal set U is a set of all elements that are relevant for some given topic of interest."
"The set theoretic difference of two sets A and B is a set of all elements in A that aren't in B."
"The complement of the complement of a set returns the original set."
"The De Morgan's laws say that the complement of A union B is equal to A complement intersect B complement."
"The power set contains all subsets of a given set."
"Are there any problems with trying to describe everything in terms of set theory? That is gonna be the topic of a different video."
"Proper subset: A is a proper subset of B if B contains something extra not in A."
"The set of the natural numbers and the set of all integers agree in cardinality."
"The cardinality of the real numbers is greater."
"The cardinality of the power set of a must be strictly greater than the cardinality of the original set."
"The cardinality of a set is the number of elements in it."
"A place there must be where union between the two sets is possible."
"The supremum of a set is the least upper bound of that set."
"An infinite set has a strange feature: it can be put into a one-to-one correspondence with a subset of itself."
"A relation is any set of ordered pairs."
"The union of two sets creates a set that contains all values in either set."
"An intersection... will contain only values in both sets."
"The power set for S is the set of all subsets of S, and it's denoted with a capital P."
"If we have a set of all sets that do not contain themselves, does that set contain itself?"
"Set theory is just basically a branch of mathematics that deals with sets and their properties."
"A subset is a set that is part of another set."
"A superset is a set that is made up of another set."
"The number of subsets is 2 to the power n, where n denotes the number of elements."
"The intersection of A and B deals with common elements between or among the sets under discussion."
"Commutative laws under set theory: A union B is the same as B union A."
"Associative laws help us understand how grouping affects the combination of sets."
"The union of a set with its complement basically gives you the universal set."
"A complement of a set is made up of elements outside that given set but within the universal set."
"The intersection of A and B... is the region that is common between A and B."
"A Singleton set is basically a set with a single member or single element."
"A power set is a set that consists of all the subsets of a given set."
"One way to show that a set is closed is to show that the complement is open."
"What a lovely set theory puzzle that took some spotting."
"If A is contained in B, then the probability of A is less than or equal to the probability of B."
"You just count how many things are in the set and that's the cardinality."
"Think of a set as a cardboard box and the elements are the things that you see when you look into the box."
"The cardinality of the empty set is zero because there is nothing in it."
"The probability of A or B is the probability of A plus the probability of B minus the probability of A and B."
"An abstract simplicial complex is a collection of sets that have certain properties."
"We have a set of real numbers... all the fractions, all the decimals, all the integers we could have."
"The elements in C, including those that are common in A and B, must not be part of the answer."
"Every non-empty subset of the real numbers which is bounded above has a least upper bound."
"The epigraph of a convex function is a convex set."
"For a given involution on a set, each element must either map to itself or be paired with a different element."
"I'm the classical nerd and today we're talking about set theory, a crazy world where minor and major chords are exactly the same."
"Set theory is the practice of dividing music, predominantly atonal music, up into chunks of three, four, or six notes which are then known as sets."
"Understanding set theory is not just a critical component of analyzing much atonal music, but it's a critical component of appreciating this style of music as well."
"If A and B are groups, then there is a natural group structure on the set A cross B."
"A union B either A happened or B happened, one or the other, or both."
"This common part is called as A intersection B."
"The union of the two sets includes all of A, the overlap between A and B, and it includes B."
"A metric space is a set X with a function D, which we will call our metric."
"Set theory, a powerful tool for describing different intervalic relations between a wide range of different types of pitch collections."
"A sublevel set is the set of all input points X for which the function is at most C."
"If you partition a set, and you want its probability, then you can just add up all the pieces."
"Let's let B sub X be n sub 1 of X, this is the ball of radius one."
"The concept of compactness is an intrinsic notion of the set and doesn't matter what metric space you're in."
"A set is connected if it's impossible to find two separated nonempty sets whose union equals to the whole set."
"A set is bounded if it stops going out toward infinity."
"The Cantor set... a very interesting subset of the real line."
"Cantor set... can be covered for any epsilon, for any positive number epsilon, can be covered with sets of size less than epsilon."
"A set E is connected if E is not the union of two separated sets."
"The countable union of countable sets is in fact countable."
"Compact sets are the next best thing to being finite."
"A set E is open if every point of E is an interior point of E."
"A set is closed if it contains all its limit points."
"A set is dense in X if its closure is all of X."
"The arbitrary intersection of closed sets is closed."
"If I wanted to be a little bit more mathematical, I could say it's like the cardinality of the set of pixels such that the pixel value is equal to \( D \)."
"This symbol here means union; it just means or. It can be in A, or it can be in B, or it can be in both."
"Closure property is telling us if you take any two elements from this set, then if you operate under this operator, then that must belong to the set again."
"Major advancements in mathematics were made due to his ideas and contributions in set theory."
"An optimal solution for the set must include the optimal solution for the subproblems, displaying optimal substructure."
"A partition of a set S is a collection of disjoint non-empty subsets that have S as their union."
"An equivalence class is essentially just a grouping or a set of all elements that are related through the relation."
"This is the number of subsets of an N element set."
"A partition of A is a collection of disjoint, non-empty sets whose union is A."
"A Cartesian product of countable sets is countable."
"A set is open if and only if the complement is closed."
"Some proper subset of an infinite series can have the same cardinality as the entire series."
"A set is countable if its elements can be put in one-to-one correspondence with the set of natural numbers."
"In mathematics, to have an arithmetic, one needs more than a collection of objects."
"A topological space can be built from any arbitrary set that you have."
"A relation is defined to be a subset of the domain of A cross B."
"Cantor's idea was to extend the concept of equivalence to infinite sets in order to define an arithmetic of infinities."
"A set is closed if and only if it contains all of its boundary points."
"The cardinality of a set is what we call the number of elements in the set."
"Two sets have the same cardinality if and only if there exists a bijection between them."
"The cardinality of the natural numbers was infinite, but the cardinality of the real numbers was an even larger infinity."
"We saw how to combine sets using operations like intersection, union, difference, and symmetric difference."
"We can't have a set of all sets as it would lead to Russell's paradox."
"Sets are collections of things; what are things? They're nouns."
"The set of all X for which P of X is true is called the truth set of P."
"This is the inclusion-exclusion theorem in the case of two sets."
"A set is just a collection of objects."
"The set of all X's such that X is in between negative 3 and 1 and includes negative 3."