MATHEMATICS
Resources & Insights
Explore mathematics, the foundational science of numbers, patterns, and logical reasoning. Learn key topics including algebra, geometry, calculus, statistics, and number theory. Ideal for students, educators, and enthusiasts seeking to develop problem-solving skills and apply mathematical concepts in science, technology, finance, and everyday life.
Equivalent Sets
Two sets A and B are said to be equivalent(≡) if each element of A is also an element of B and each element of B is also an element of A. If elements are repetitive in one set, then it is not required for it to repeat in the other set for the two sets to be equivalent. Examples ⇒ { 1, 2, 3 } ≡ { 1, 3, 2} { 1, 2, 3 } ≡ { 1, 3, 2, 2, 1, 2, 3, 3 } { 1, 2, 3 } ≢ { 1, 3, 2, 2, 1, 2, 3, 3, 4 } Since 4 is not in first set READ MORE...
Equal Sets
Two sets A and B are said to be equal(=) if they have the same elements. The elements may not be in the same order. If an element appears n times in one set, then it must also appear n times in the other set. Mathematically, two sets A and B are equal if For each a ∈ A, there exists an element b ∈ B. n(A) = n(B) i.e. cardinality of A is equal to cardinality of B. Examples ⇒ { 1, 2, 3, 4 } = { 1, 3, 4, 2 } { 1, 1, 2, 2, 2, 4, 5 } = { 1, 4, 1, 2, 5, 2, 2 } { 1 ,2 ,3 ,4 } ≠ { […] READ MORE...
Finite Set
A set that has a definite number of elements is called a finite set else it is called an Infinite set. ⇒ A null set is a finite set. ⇒ For finite set S, n(S) is a finite number. ⇒ The standard mathematical sets like N, Z, R, etc. are all infinite. READ MORE...
Singleton Set
A set that has exactly one member is called a singleton set. { 1 } , { ‘a’ } , { x3 | x ∈ N , 2 < x < 3 } are all singleton sets. READ MORE...
Empty Set
A set that does not have any members is called an empty set. ⇒ Such sets are represented as {} or using the symbol ∅ (derived from Φ) which has been explicitly developed to designate an empty set. ⇒ Sometimes { ∅ } is also used to designate an empty set. ⇒ An empty set is also called a Void set. ⇒ The cardinality of an empty set is 0. READ MORE...
Cardinality of Sets
The number of elements in a set is called cardinality. The cardinality of a set A is generally represented by |A| or n(A) meaning the number of elements in set A. Examples: The cardinality of the set { 1, 45, 2, 34 } is 4. The cardinality of the set { a, e, i, o, u } is 5. The cardinality of the set { x | x ∈ N } is infinite ( a very large number whose value cannot be determined ) READ MORE...
Element Position in Sets
The position of elements in a set does not change the value or the meaning of the set. The above statement signifies that { 1, 2, 3 } and { 1, 3, 2 } and { 2, 3, 1 } are all the same set. A set is primarily a collection and not a sequential representation of elements. It represents a group as a whole. The {} representation just shows the group in an expanded form. By re-arranging the members of a set, you do not change the group. Just like rearranging the seating position of students makes no changes to the class as a whole (it still contains the same students) or transferring books from one shelf to another […] READ MORE...
Set Membership
As defined earlier, a set is a collection or group of objects. These objects are called members of the Set. This relationship is represented by using the symbol ∈. The symbol ‘∈’ means “is a member of ” or “belongs to” or “is an element of”. The reverse of this relationship is denoted using the symbol ‘ ∋‘. ∋ means “contains as a member“. If V represents the collection of all vowels in the English language, then ‘a’, ‘e’, ‘I’, ‘o’, ‘u’ are all members of this set. We can, hence, write, ‘a’ ∈ V ‘e’ ∈ V ‘i’ ∈ V ‘o’ ∈ V ‘u’ ∈ V We can also write V ∋ ‘u’ Similarly, ∉ and ∌ are used to denote the negation of the membership. ∉ means “is […] READ MORE...