Table of Contents
A set is indeed a grouping of well-defined objects. Only on the basis of simplicity are the objects of a set considered distinct.
A family or collection of sets is yet another name for a set of sets. Assume we have a family of sets consisting of A1, A2, A3,….. up to An, which is the family {A1, A2, A3,….., An } and could be denoted as:
S = {Ai | i belongs to N and 1 ≤ i ≤ n}
Types of Sets
Singleton set: The set which only has one element.
Empty Set/Null Set: An empty set is a collection that contains no elements.
Proper set: If indeed A and B are two sets, then A is said to be a proper subset of B if A ⊆ B but A ≠ B.
Power Set: The power set of a set is really the collection of all its subsets. If A is indeed the set, then P(A) is its power set.
The number of items in any power set can be calculated using n[P(A)] = 2n, where n is the number of elements in set A.
Relations
The relation would be useful for determining the relationship between a function’s input and output.
A non-empty set P to another non-empty set Q relation R is a subset of P X Q.
Types of Relations
Empty Relation: An empty relation (or void relation) would be one in which no elements of a set are related to one another.
Universal Relation: A universal (or full) relation would be one in which every element of a set is related to each other.
Identity Relation: In quite an identity relation, every element of a set is only related to itself.
Inverse Relation: When a set contains elements that are inverse pairs of elements from another set, this is referred to as an inverse relation.
Reflexive Relation: Each element in a reflexive relationship maps to itself.
Symmetric Relation: When a=b is true, then b=a is also true in an asymmetric relationship.
Transitive Relation: For this relation, when (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R.
Equivalence Relation: An equivalence relation is something that is reflexive, symmetric, and transitive all at the same time.
Functions
A function has been simply used to represent the dependence of one quantity on another and is easily defined using the mapping concept. In basic terms, a function is a relation that produces one output for each input.
One function from set P to set Q is indeed a rule that assigns one and only one element of set Q to each element of set P.
Domain, Co-domain and Range of a Function
A function is a mathematical relation between two sets, usually denoted by an equation. The function assigns a unique output to every input. The input and output can be real numbers, but they can also be complex numbers.
The domain of a function is the set of all inputs for which the function produces a result. The co-domain is the set of all outputs for which the function produces a result. The range is the set of all outputs that the function can produce.
For example, the domain of the function f(x) = x2 is all real numbers. The co-domain is all real numbers. The range is all real numbers except for negative numbers.
The domain of the function g(x) = x3 is all real numbers. The co-domain is all real numbers. The range is all real numbers except for negative numbers and zero.
The domain of the function h(x) = x is all real numbers. The co-domain is all real numbers. The range is all real numbers except for zero.
FAQs
What is set relation?
A relation between two sets would be a collection of ordered pairs that each contains one object from the other set.
How do you tell if a set of relations is a function?
When a set of ordered pairs passes the vertical line test, it is a function. Since there is only one corresponding value for any given value, the ordered pair relation IS a function.
