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.
Complement of a set
The complement of a set A (that is a subset of a universal set U) is defined as a set that contains all the member elements and all subsets of U that are not part of the set A. The complement of a set is denoted using the symbol ‘ or c. Hence, A’ = Ac = complement of A Example:- If N is the universal set, and if A = { 1, 2,3 } then A’ = { x ∈ N | x >3 } A = { set of all boys in a class }, then A’ ={ set of all girls in a class } A = { set of all vowels in English language }, then A’ = { set of all consonants […] READ MORE...
Venn Diagrams in Sets
Venn diagrams are the pictorial or graphical representation of sets and the various relationships that exist between sets. The representation consists of a rectangular box representing the universal set(U). All sets that are in context are drawn as circles and within the area of the rectangular box. These diagrams were devised by John Venn and hence they are named after him. A typical layout of a Venn diagram is shown below. READ MORE...
UNIVERSAL SET
When working with sets, a reference superset that contains all the sets and their subsets in context (i.e. the sets which are currently under consideration) is called a universal set. It provides a base set from which all of the sets of interest can be derived or deduced. The universal set concept is needed at times in solving computational problems related to sets and providing certain axioms and in the application of Venn diagrams. When dealing with multiple collections of the same objects, then a set that contains all the possible objects and all sets and subsets formed from these elements is called a superset of every other set or universal set. It is generally denoted by the U. There are no laws […] READ MORE...
Partitions
Partitioning of a set is distributing the member elements of a set among a group of non-empty subsets in such a way that each member lies in only one of these subsets. ⇒ ∅ ( Empty Set ) cannot be the partition of any set. Examples ⇒ The set { 1, 2, 3 } can be partitioned in the below subsets :{ 1 } , { 2, 3 }{ 2 } , { 1, 3 }{ 3 } , { 1, 2 }{1}, {2}, {3} READ MORE...
Disjoint Sets
Two sets A and B are called disjoint when they have no element in common (except the empty set ∅ }. READ MORE...
Power Set
The set of all possible subsets of a set S is called the power set of S, written as P(S). Examples ⇒ The power set of { ‘a’ } is { ∅, {‘a’} }The power set of { 2, 3 } is { ∅, {2}, {3}, {2,3} }The power set of {1, 2, 3 } is { ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } If there are n members in a set S i.e. cardinality of set |S| = n, then there are 2n subsets possible.Hence the power set would contain 2n elements. Or we can say that the cardinality of the power set is |P| is 2n. READ MORE...
Proper Subset
If A is a subset of B but A ≠ B, then A is called the proper subset of B, and B is called the proper superset of A. This relationship is represented as below A ⊂ B ( A is a proper subset of A )B ⊃ A ( B is a proper superset of A ) Examples ⇒ { x2 | x ∈ N } ⊂ N ( not all natural numbers are squares ){ 1, 2, 3 } ⊂ { 1, 2, 3, 4 }{ 1, 2, 3, 3 } ⊂ { 1, 2, 3, 4 } READ MORE...
Subset
If there are two sets A & B such that every element of A is also in B, then A is called a subset of B. In other words, A is contained in B. B is called the superset of A. In the set theory, this relationship is depicted as below A ⊆ B ( A is a subset of B) B ⊇ A ( B is a superset of A ) ⇒ ∅ (empty set) is the subset of every set ⇒ A set S is a subset of itself. Examples ⇒ { 1 } ⊆ { 1, 2, 3 } ∅ ⊆ { 1, 2, 3 } { 1, 2, 3 } ⊆ { 1, 2, 3 } { x2 | x ∈ N […] READ MORE...