Let's break down the concepts of sets, relations, and functions with examples in JavaScript.
Sets, Relations, and Functions:
1. Sets:
A set is a collection of well-defined objects or elements. These objects can be anything like numbers, letters, or even other sets. Sets are usually denoted by curly braces {}.
Example of a Set:
Operations on Sets:
Union (A ∪ B): Combines elements from both sets.
Intersection (A ∩ B): Finds elements common to both sets.
Difference (A - B): Elements in A that are not in B.
2. Relations:
A relation between two sets is a collection of ordered pairs where the first element is from the first set, and the second element is from the second set. In JavaScript, a relation can be represented as a set of pairs.
Example of a Relation:
Consider two sets:
- Set A = {1, 2, 3}
 - Set B = {a, b, c}
 
A relation R from A to B can be represented as a set of ordered pairs:
Types of Relations:
Reflexive Relation: A relation R is reflexive if every element of set A is related to itself. For example, if A = {1, 2, 3}, then R = {(1, 1), (2, 2), (3, 3)} is reflexive.
Symmetric Relation: A relation R is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R. Example: If (1, 2) is in R, then (2, 1) must also be in R.
Transitive Relation: A relation R is transitive if for all a, b, c ∈ A, whenever (a, b) and (b, c) are in R, then (a, c) must also be in R.
3. Functions:
A function is a special type of relation where each element in the domain (the first set) is related to exactly one element in the codomain (the second set). A function can be denoted as f: A → B, where A is the domain, and B is the codomain.
Example of a Function:
Let’s say:
- Set A = {1, 2, 3}
 - Set B = {a, b, c}
 
A function f: A → B could be:
Types of Functions:
One-to-One (Injective): A function is one-to-one if different elements of the domain map to different elements of the codomain. In our example, f is one-to-one because no two elements in A map to the same element in B.
Onto (Surjective): A function is onto if every element of the codomain has at least one element from the domain mapping to it. For instance, if B = {a, b, c} and f = {1 → a, 2 → b, 3 → c}, the function is onto because every element in B has a corresponding element in A.
One-to-One Correspondence (Bijective): A function is bijective if it is both one-to-one and onto, meaning there is a perfect pairing between the elements of the domain and codomain.
Example of a Bijective Function:
Let’s say:
- Set A = {1, 2, 3}
 - Set B = {a, b, c}
 
A bijective function f: A → B could be:
In this case, the function is both injective and surjective, making it bijective.
Summary of Examples:
- Set Operations (Union, Intersection, Difference)
 - Relation (Set of ordered pairs)
 - Function (Mapping of domain to codomain)
 
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