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Operations in various Mathematical Structures

Operations in various Mathematical Structures

Photo Courtesy: https://blogs.scientificamerican.com/

Mathematics and Statistics are two of the most important concepts of Data Science. Data Science revolves around these two fields and draws their concepts to operate on the data.

Today, we will go through the various operations in different types of Mathematical structures.

The learners of Mathematics so far have the awareness about the below mathematical structures:

  • Real number system
  • Complex number system
  • Vector Spaces
  • Matrix Algebra
  • Set Theory
  • Group Theory
  • Graph Theory
  • Fields and Rings
  • Number theory etc.

Every mathematical structure mentioned above has its unique identity in the study of Mathematics. Moreover every structure has certain basic motivation behind it. Every theory has its own set of notations, operations and limitations. All these theories have some of the interesting applications in computer science. Just to quote few Real number system forms the basis for all computations. Vectors spaces and Matrix algebra is essential in ML, AI and other studies. Graph theory plays important role in network theory.

In the present blog an attempt is made to summarize the notations,
operations and difference between the set of operations.

Real Number System: It is denoted by R, graphically it is a real line or X axis.

  • The base of real number system is {0,1,2,3,4,5,6,7,8,9}, so it is called as decimal number system.
  • The different alphabets are used with different meaning e.g.
    • Letters a,b,c….. are used denote unknown constants & elements of Set
    • Letters i,j,k…… are used as counters in matrix algebra.
    • Letters u,v,w…. are used denote vectors in vector spaces.
    • Letters X,Y,Z…..are used to denote variables.
    • Letters A,B,C….are used to denote the Matrix or Set.

  • Arithmetic Operations: The set of operators are
    • + (Addition), – (Subtraction) * (Multiplication) and / (Division)
    • The power operation is the composite operation by using *
    • All operations leads to a real number and this property is called as closure property.
    • A valid combination of constants, variables and arithmetic operators is known as arithmetic expression.

  • Relational Operators: These are
    • Less than <, More than >, Equal to =, Not equal to ≠, Less than or equal to ≤, more than or equal to ≥.
    • These are operators used for numerical comparison and hence leads to either True (1) or False (0)
    • A valid combination of constants, variables, arithmetic expressions and relational operator is known as logical expression.

  • Vector Spaces: It can be considered as a slight extension of complex number system. These are specially known for directional studies. It is denoted by Vector space V with left headed arrow on top.
    • The basis of vector space of ‘n’ dimensional vector space includes ‘n’ unit vectors along each direction of ‘n’ axis and it is {(1,0,…,0), (0,1,0,…,0)…(0,0,…,1)}. One can think that these are rows or columns of identity matrix of order n X n.
    • The vector space is denoted by U or V or W. The elements of vector space is denoted by {(u 1 i, u 2 j, u 3 k)/ u 1 , u 2 , u 3 are real numbers, i, j, k are unit vectors along X, Y and Z axis}. The concept can be extended for ‘n’ dimensional vector space.
    • The operations in vector space includes + (Addition), – (Subtraction), scalar product and vector product. Addition and substation are the point-wise additions. The scalar product means each component of vector is multiplied by the same constant. The vector product is another special feature of vector spaces and it takes care of change in direction while performing product.
    • There is NO concept like vector division or inverse of a vector.
    • These Vector Spaces are useful to understand the concept like reflection, transformation, displacement, contraction and expansion. One can correlate it with Zoom in or Zoom out features.

  • Matrix Algebra: In combination with vector space it is also called as Linear Algebra. It is the ordered arrangement of element in rows and columns.
    • Matrix is denoted by A, B, C…
    • Every matrix is known by its order, means number of rows and columns. Rows and columns are usually denoted by n and m.
    • A matrix can also be denoted by A= ((a ij )) i=1,2,..n and j=1,2,..m
    • The letters a,b,c… are used to show the element and letters i, j are used to indicate the position in the matrix.
    • The matrix operations are + (Addition) – (Subtraction). These operations are not in general, rather they are suitable only if both matrices have same order. It means addition and subtraction is NOT on any two matrices.
    • Transpose of a matrix one of the special operation, it gives a matrix on interchange of rows and columns.
    • Scalar multiplication. It is also possible on every matrix. It is just multiplying each element of the matrix by same constant.
    • Matrix multiplication is a special feature in matrix theory. It is defined NOT in general. Multiplication of two matrices A and B is AB valid is and only if number of columns of matrix A are same as number of rows of matrix B. The resulting matrix have still different order. Here AB and BA may NOT be same, means the commutative property is not satisfied by matrix multiplication.
    • Matrix division is NOT defined. Instead the concept of Matrix inverse is defined. This is also a very specialized concept. It is valid only for square matrices with non-zero determinant (Non-singular Matrix). Yet for other matrices there is a concept of generalized inverse and Moore Penrose G inverse.
    • Matrix theory in general is very useful in solving a system of equations more over in AI and ML, it plays important role.

  • Set Theory: It is one of the different kind of mathematical structure with altogether different concept. It is based on the simple idea of collection of objects. These objects may not be necessarily of same type but must be rigidly defined.
    • The letter A, B, C,… are used for a set. The letters a, b, c,… are used denote the elements of a set.
    • The basic operations in set theory are Union, Intersection and complement. Union ‘U’ is the combing the objects in two or more sets without repetition. Intersection ‘∩’ means collecting the common objects in two or more sets. The operation of complement refers to collecting the objects from parent set but not in given set.
    • There are NO concept like formal addition, substation, multiplication and division or inverse in set theory.
    • Union, Intersection and Complement defines further operations like subtraction of two sets and symmetric difference between two sets.
    • It is necessary in the probability study and fuzzy mathematics.
    • Many software do not provide the features for set theoretic operations but open source software R provides Union and Intersection operation.

Thus in conclusion, it is very important to understand the difference between the set operations in various mathematical structure.

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