Set theory is a fundamental branch of mathematics that deals with the study of sets, which are collections of distinct objects called elements. It includes various notations and symbols for defining and manipulating sets, such as the universal set, cardinality, empty set, and set membership. The theory also covers operations like union, intersection, and Cartesian product, and adheres to laws like commutative, associative, and distributive. Understanding set theory is crucial for organizing and analyzing data systematically.
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Sets are well-defined collections of distinct objects, enclosed within curly brackets
Universal set (U)
The universal set contains all elements under consideration in a particular context
Cardinality (|X|)
The cardinality of a set denotes the number of elements in the set
Empty set (∅)
The empty set signifies a set with no elements
Elements
Each set is characterized by its elements, which must be clearly defined and distinct
Cardinality
The cardinality of a set counts the unique elements within the set
Sets can be defined by a clear property, such as "the set of all even numbers less than 10."
Sets can be represented by listing out their elements, such as {2, 3, 5, 7} for the set of prime numbers less than 10
Sets can be expressed by specifying a rule or condition that its members must satisfy, such as {x | x is a prime number, x < 10}
Null set (∅)
The null set contains no elements
Singleton set
A singleton set has precisely one element
Finite and infinite sets
Finite sets have a limited number of elements, while infinite sets do not have a finite cardinality
Equal sets
Equal sets possess exactly the same elements
Equivalent sets
Equivalent sets have equal cardinalities, though their elements may differ
Disjoint sets
Disjoint sets share no common elements
Subset
A subset contains all elements of another set
Superset
A superset contains all elements of another set, with a proper subset being strictly smaller
The union of sets merges all elements from the sets, eliminating duplicates
The intersection of sets finds elements common to both sets
The complement of a set consists of elements in the universal set but not in the set
The Cartesian product creates ordered pairs from the elements of two sets
The set difference isolates elements in one set that are not in another
00
______ is a crucial mathematical field that deals with collections of unique items called elements.
Set theory
01
The term ______ refers to the number of distinct elements in a set, and the symbols (∈) and (∉) represent ______ and ______ to a set, respectively.
cardinality
membership
non-membership
02
Verbal description form of sets
Defines a set by stating a clear property, e.g., 'all even numbers < 10'.
