Distinct Elements: Each element in a set is unique, meaning that no two elements in the set are the same. If an element appears more than once, it is still counted only once in the set.
Well-Defined: A set must have clear and unambiguous criteria for determining whether an element belongs to it or not. This means you should be able to determine whether any given object is a member of the set or not without any ambiguity.
Unordered: Sets are generally considered unordered collections, which means that the elements in a set do not have a specific order. The order in which you list the elements of a set does not affect the set itself.
Sets are typically represented using curly braces {} and a list of their elements.
For example, here is a set containing the first three positive integers:
{1, 2, 3}
If you want to represent an empty set, you can use {} or ∅ (the symbol for an empty set).
A is comparable to B if and only if A ⊆ B or B ⊆ A
Subset
A subset is a set that contains only elements that are also in another set. If every element of set A is also an element of set B, then A is considered a subset of B. This relationship is denoted as A ⊆ B.
Proper Set
A proper set is a set that is not equal to its own subset. In other words, a set A is considered proper if there exists another set B such that every element of A is also an element of B, but A is not equal to B.
Finite Set
A finite set is a set that contains a finite (limited) number of elements. For example, the set {1, 2, 3} is finite because it has three elements.
Infinite Set
An infinite set is a set that contains an infinite number of elements. A classic example of an infinite set is the set of all natural numbers {1, 2, 3, ...}.
Singleton Set
A singleton set is a set that contains exactly one element. For example, {5} is a singleton set because it contains only the element 5.
Equal Set
Two sets are equal if they have exactly the same elements. In set notation, if every element of set A is also an element of set B, and vice versa, then A = B.
Equivalent Set
Equivalent sets are sets that have the same cardinality, meaning they contain the same number of elements. Two sets can be equivalent even if their elements are different. For example, the set {1, 2, 3} and the set {4, 5, 6} are equivalent because both have three elements.
Universal Set
The universal set, often denoted as U, is the set that contains all the elements under consideration in a particular context. It is a way to define the largest set within a given context.
Power Set
The power set of a set A, denoted as P(A), is the set of all possible subsets of A, including the empty set and A itself. If A has n elements, its power set will have 2^n subsets.
Index Set
An index set is a set used to index or label other sets, often in the context of sequences or families of sets. For example, the set {1, 2, 3} can be indexed by the index set {1, 2, 3} to represent a sequence.
Cartesian product
The Cartesian product of two sets, often denoted by A × B, is a mathematical operation that produces a new set containing all possible ordered pairs where the first element comes from set A and the second element comes from set B.
Mathematically, if A = {a₁, a₂, a₃, ...} and B = {b₁, b₂, b₃, ...} are two sets, then their Cartesian product A × B is defined as:
A × B = {(a₁, b₁), (a₁, b₂), (a₁, b₃), ...,
(a₂, b₁), (a₂, b₂), (a₂, b₃), ...,
(a₃, b₁), (a₃, b₂), (a₃, b₃), ...,
In other words, the Cartesian product contains all possible combinations of one element from set A paired with one element from set B, arranged in ordered pairs.
Here's a simple example: Let A = {1, 2} and B = {x, y}. The Cartesian product A × B would be:
A × B = {(1, x), (1, y), (2, x), (2, y)}
Relation
A relation between two sets, often denoted as R, is a set of ordered pairs, where the first element of each pair belongs to the first set (the domain), and the second element belongs to the second set (the codomain).
In formal terms, if A and B are sets, a relation R from A to B is defined as a subset of the Cartesian product A × B. It consists of pairs (a, b) such that a is an element of A, and b is an element of B.
Relations can be represented as sets of ordered pairs. For example, consider the relation "is less than" between the sets A = {1, 2, 3} and B = {4, 5, 6}. It can be represented as R = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (3, 6)}.
Relations can have various properties, such as being reflexive, symmetric, transitive, or antisymmetric, which depend on the specific relationship they represent.
Reflexive Relation
A relation R on a set A is reflexive if every element of A is related to itself. In other words, for all elements a in A, (a, a) must be in the relation R.
Symmetric Relation
A relation R on a set A is symmetric if for all elements a and b in A, if (a, b) is in R, then (b, a) must also be in R.
Transitive Relation
A relation R on a set A is transitive if for all elements a, b, and c in A, if (a, b) is in R and (b, c) is in R, then (a, c) must also be in R.
Antisymmetric Relation
A relation R on a set A is antisymmetric if for all distinct elements a and b in A, if (a, b) is in R, then (b, a) cannot be in R.
Irreflexive Relation
A relation R on a set A is irreflexive if no element of A is related to itself. In other words, for all elements a in A, (a, a) cannot be in the relation R.
Equivalence Relation
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint subsets (equivalence classes) such that elements within the same subset are related to each other in the same way.
Partial Order (Partial Ordering Relation)
A partial order relation is a relation that is reflexive, antisymmetric, and transitive. It is used to describe partial orders or hierarchies, where some elements are related by "less than or equal to" or "precedes" relationships.
Equivalence Relation on a Partition
Given a partition of a set A into disjoint subsets, an equivalence relation can be defined such that two elements are related if and only if they belong to the same subset in the partition.
Binary Operation
A binary operation is a specific rule or function that takes two elements from the set and combines them to produce another element from the same set. Mathematically, if "∗" is a binary operation on a set S, then for any two elements a and b in S, the operation is written as "a ∗ b," and the result is also an element of S.
Function
A function is a special type of relation in which each element from the domain (the first set) is associated with exactly one element in the codomain (the second set).
Formally, a function f from set A to set B is a relation such that for every element a in set A, there exists a unique element b in set B such that (a, b) is in the relation. The element a is called the input or argument of the function, and b is the output or value of the function for that input. The notation f(a) represents the value of the function f for the input a. Functions can be represented using various notations, such as functional notation (e.g., f(x)), arrow notation (e.g., A → B), or as sets of ordered pairs.
For example, consider the function "square" from the set of real numbers to itself, denoted as f: ℝ → ℝ, defined by f(x) = x^2. This function assigns a unique square value to each real number as its input.
Domain of a Function
The domain of a function is the set of all possible input values (or arguments) for that function. It is the set of values over which the function is defined and can produce output.
Range of a Function
The range of a function is the set of all possible output values that the function can produce when its domain elements are used as input.
Injective (One-to-One) Mapping
An injective mapping, also known as a one-to-one mapping, is a function in which every distinct element in the domain maps to a distinct element in the codomain. In other words, no two different elements in the domain map to the same element in the codomain.
Mathematical Definition: A function f: A → B is injective if, for every pair of distinct elements a₁ and a₂ in set A, f(a₁) ≠ f(a₂).
Surjective (Onto) Mapping
A surjective mapping, also known as an onto mapping, is a function in which every element in the codomain is mapped to by at least one element in the domain. In other words, the function covers the entire codomain.
Mathematical Definition: A function f: A → B is surjective if, for every element b in set B, there exists at least one element a in set A such that f(a) = b
Bijective Mapping
A bijective mapping is a function that is both injective and surjective. In other words, it is a one-to-one and onto mapping.
Mathematical Definition: A function f: A → B is bijective if it is both injective and surjective.
Function Composition
Function composition is an operation that combines two functions to create a new function. Given two functions f: A → B and g: B → C, the composition (g ∘ f): A → C is defined as (g ∘ f) (x) = g(f(x)) for all x in A.
Identity Mapping
The identity mapping is a special function that maps each element in a set to itself. In other words, for any set A, the identity mapping is denoted as id_A: A → A, and it satisfies id_A(x) = x for all x in A. A function f: A → B has an inverse function f^(-1): B → A if, for every element b in B, there exists a unique element a in A such that f^(-1)(b) = a. In addition, f(f^(-1)(b)) = b and f^(-1) (f(a)) = a for all a in A and b in B.
A function f: A → B is a constant mapping if there exists a single element b in B such that f(a) = b for all a in A. Given two sets A and B, an equal mapping can be represented as a relation that checks whether A is equal to B or not. It returns true (or establishes a relation) if A and B are equal sets and false (or no relation) if they are not equal.
