Table of Contents
Sets in Maths
A set is 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 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}
A set is denoted by a capital letter and represented by listing all of its elements between curly brackets, such as { }.
Types of Sets
- Set of singletons
A set only has one element. For instance, A = 3 and B = pencil. Because A and B each contain only one element, they are both singleton sets.
Null Set/Empty Set
An empty set is a set that contains no elements. It is represented by A = { } or A = ϕ.
- Proper set
If A and B are two sets, then A is a proper subset of B if and only if A ⊆ B but A ≠ B. For instance, if B = {2, 3, 5} , A = {2, 5} is a proper subset of B.
- Power Set
A set’s power set is the collection of all its subsets. If A is the set, then P(A) is its power set. The number of elements in any power set can be calculated using n[P(A)] = 2n, where n is the number of elements in set A.
- Set that is finite
A set is made up of a finite number of elements.
- Infinite collection
A set is called an infinite set if the number of elements in it is infinite.
Operations on Sets
Insets theory, there are three basic operations that can be applied to two sets:
- Combination of two sets
- Two-set intersection
- Difference between two sets
Relations in Maths
The relation is 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) is one in which no elements of a set are related to one another.
- Universal Relation: A universal (or full) relation is one in which every element of a set is related to each other.
- Identity Relation: Every element of a set is only related to itself in an identity relation.
- 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: Every element in a reflexive relationship maps to itself.
Functions in Maths
A function is simply used to represent the dependence of one quantity on another and is easily defined using the mapping concept. In layman’s terms, a function is a relation that produces one output for each input. A function from set P to set Q is a rule that assigns one and only one element of set Q to each element of set P.
A function f from a set P to a set Q, denoted as f: P-> Q, is a mapping of elements of P (domain) to elements of Y (co-domain) in which each element of P is assigned to some chosen element of Q. That is, each element of P must be assigned to at least one element of Q.
Domain:
The set of all x values for a function, y = f(x), is known as the domain of the function. It refers to the set of input values that can be used.
Range: The range of y = f(x) is a collection of all f(x) outputs for each real number in the domain. The range is the collection of all y values.
FAQs
What exactly is a set relation?
A relation between two sets is a collection of ordered pairs that each contains one object from the other set. If object x belongs to the first set and object y belongs to the second set, the objects are said to be related if the ordered pair (x,y) is in the relation.
What is the fundamental relationship between sets?
Sets are well-defined collections that are entirely defined by their elements. As a result, two sets are equal if and only if they have the same elements. The fundamental relationship in set theory is that of element hood, also known as membership.
