Presentation_on_Matrices,_Fun [1].pptx

Presentation_on_Matrices,_Fun [1].pptx

1. Presentation on Matrices, Functions, and Relations

2. CONTENTS 1. History of Matrices 5. Boolean Matrices 9. Mathematicians and Their Contributions 2. Mathematicians Linked to Matrices 6. Applications of Matrices 10. Introduction to Functions 13. Related Diagrams and Mathematical Expressions 14. Summary of Functions 3. Introduction to Matrices 17. Introduction to Relation 7. Summary 11. Types of Functions 18. Graphical Representation 21. Summary of Relations 15. History of Relation 4. Matrix Operations 19. Properties of Relations 8. History of Functions 12. Related Types and Examples 16. Mathematical Linked to Relations 20. Mathematical Expressions

3. 1. History of Matrices The concept of matrices can be traced back to ancient civilizations. Around 200 BCE, Chinese mathematicians were already arranging numbers in grid-like forms to solve systems of linear equations. In the book Jiu Zhang Suàn Shù (The Nine Chapters on the Mathematical Art), problems involving linear equations were solved using matrix-like tables — one of the earliest recorded uses of matrix concepts. The modern term 'matrix' was introduced by James Joseph Sylvester in 1850. His close friend Arthur Cayley later published the first paper on matrix algebra in 1858, introducing matrix multiplication and the concept of the determinant. This development marked the beginning of modern linear algebra. The formal concept of matrices emerged in the 19th century when English mathematician Arthur Cayley introduced matrix multiplication and the idea of the identity matrix. He also developed the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic equation.

4. 2. Mathematicians Linked to Matrices Arthur Cayley (1821–1895) Developed the matrix theory and introduced the Cayley–Hamilton theorem. James Joseph Sylvester (1814–1897) Coined the term 'matrix' and laid the foundation of matrix algebra. Charles Hermite and Ferdinand Frobenius Worked on determinants, eigenvalues, and canonical forms of matrices. William Rowan Hamilton Invented quaternions, which influenced matrix representation of rotations. Gottfried Wilhelm Leibniz (1646–1716) One of the first to study determinants while working on systems of linear equations, paving the way for future matrix theory.

5. 3. Introduction to Matrices A matrix is a rectangular array of numbers arranged in rows and columns. It provides a compact way to represent and work with sets of equations or data. Each item in a matrix is called an element or entry and is usually denoted as aij, where 'i' is the row number and 'j' is the column number. Example of a 2×3 matrix: Here, A has 2 rows and 3 columns. The element a12 = 2 (1st row, 2nd column). Matrices can contain integers, real numbers, or even symbols depending on the problem. Different types of matrices include: 1 Row Matrix Only one row exists. 3 Square Matrix Same number of rows and columns. 5 Diagonal Matrix 2 Column Matrix Non-zero elements appear only on the main diagonal. Only one column exists. 4 Zero Matrix 7 Every element is zero. Upper and Lower Triangular Matrices 6 Only elements above or below the diagonal are nonzero. Identity Matrix (I) A diagonal matrix with ones on the main diagonal.

6. 4. Matrix Operations Matrices can be manipulated using several operations: Addition Two matrices can be added only if they have the same dimensions. Example: A + B = [aij + bij]. Scalar Multiplication Each element of a matrix is multiplied by a constant (kA = [k·aij]). Transpose (AT) Rows and columns are interchanged. Matrix Multiplication Determinant and Inverse If A is m×n and B is n×p, their product C = AB is m×p, given by: cij = Σ (aik × bkj). For square matrices, determinants help solve equations and find inverses.

7. 5. Boolean Matrices Boolean matrices are special matrices that use only 0s and 1s as elements. They are very useful in computer science, especially for representing relations, logic circuits, and graphs. Logical OR (∨) Works like addition; the result is 1 if either of the values is 1. Logical AND (∧) Works like multiplication; the result is 1 only if both are 1. Boolean Product Combines matrices using logical operations. Boolean matrices are commonly used to find transitive closures of relations or to compute the reachability of nodes in a directed graph. 1 2 3

8. 6. Applications of Matrices Matrices are an essential tool in mathematics, computer science, and engineering. They allow large sets of data or equations to be represented compactly and processed efficiently. Here are a few major applications: Computer Graphics Used to perform transformations such as rotation, scaling, and translation. Data Representation Applied in databases and image processing. Network Analysis Boolean matrices represent communication or link systems. Engineering and Physics Help solve systems of linear equations, calculate forces, analyze electrical circuits, and model physical phenomena. Graph Theory Represented using adjacency matrices to show connections between nodes. Cryptography Used in encoding and decoding secret messages. Medicine and Biology Used in genetics, population studies, and imaging technologies like CT scans and MRI. Economics and Business Input-output models of economies use matrices to show how different sectors depend on one another for resources and output.

9. 7. Summary In conclusion, matrices are not just mathematical tools but powerful systems of organization and transformation that play a vital role in modern science, engineering, and technology. Matrices provide a structured and efficient way to handle complex numerical or logical data. Understanding their types, operations, and Boolean extensions is crucial for deeper study in mathematics.

10. 8. History of Functions The concept of functions can be traced back to ancient Greek mathematicians, particularly Euclidean Diophantus. The formal definition of a function began with the works of 17th and 18th-century mathematicians, including René Descartes and Gottfried Wilhelm Leibniz, who connected the concept of functions with geometry and calculus. The term "function" was formalized in the 19th century by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass, who developed the notion of a function as a relationship between two sets.

11. 9. Mathematicians and Their Contributions 1 Gottfried Leibniz (1646-1716) Introduced the term "function", developed calculus notation, pioneered functional relationships in curves. 3 Peter Dirichlet (1805-1859) Provided rigorous modern definition as arbitrary correspondences between sets. 2 5 Isaac Newton (1643-1727) Leonhard Euler (1707-1783) Created f(x) notation and treated functions as primary mathematical objects. Developed calculus using functions applied to physical phenomena. 4 Georg Cantor (1845-1918) Extended function theory through set theory, establishing foundations for infinite sets.

12. 10. Introduction to Functions In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function, and the set Y is the codomain, where each input is related to exactly one output. This is written as: 𝑓: 𝐴 → 𝐵 and 𝑓(𝑥) = 𝑦, where: x = input (from set A) y = output (from set B)

13. 11. Types of Functions 1 Injective (One-to-One) Each element of the domain maps to a unique element in the codomain. 3 Bijective A function that is both injective and surjective. 5 Identity Functions Surjective (Onto) 2 Every element of the codomain has at least one element from the domain mapping to it. A function where each element maps to itself. 4 Constant Functions A function where the output is the same for all inputs.

14. 12. Related Types and Examples Linear Functions Functions of the form f(x) = ax + b, representing a linear relationship. Quadratic Functions Functions of the form f(x) = ax² + bx + c, represented graphically as a parabola. Exponential Functions Functions of the form f(x) = ax showing exponential growth or decay. Logarithmic Functions The inverse of exponential functions, f(x) = log(x). Trigonometric Functions Functions like sine, cosine, and tangent, used to describe periodic phenomena.

15. 13. Related Diagrams and Mathematical Expressions Graph of Functions A visual representation of a relationship between input and output values, plotted as ordered pairs (x, y) on a coordinate plane. Linear Functions A linear function represents a straight-line graph with a constant rate of change. Its general form is y = mx + c. Quadratic Functions A quadratic function forms a parabolic curve opening upward or downward, generally expressed as y = ax² + bx + c. Exponential Functions An exponential function shows rapid growth or decay, expressed as y = a·bˣ where b is the base. Trigonometric Functions Functions like sine and cosine represent periodic wave patterns, widely used to model cycles and oscillations.

16. 14. Summary of Functions Functions are mathematical relationships where each input has a single output. The concept dates back to Greek mathematicians and was formalized by Leibniz, Euler, and others. Functions can be linear, quadratic, exponential, logarithmic, or trigonometric, each showing a unique relationship between variables. Graphs of functions visually represent these relationships on a coordinate plane, forming the foundation of modern mathematics.

17. 15. History of Relation Aristotle (338-322 BCE) Studied relations in the context of logic and categorization. George Boole (1815-1864) 1 Developed Boolean algebra, laying the foundation for modern relational algebra. 3 Charles Sanders Peirce (1839-1914) Contributed to the development of relation algebra and introduced the concept of relations as fundamental to mathematics. 5 Medieval Logician (12th-14th centuries) Developed theories of relations, including equivalence. Alfred Tarski (1901-1983) Augustus De Morgan (1806-1871) 7 Introduced relational algebra and the concept of relations as sets of ordered pairs. Worked on the theory of relations and its applications to logic and mathematics. Ernst Schroder (1841-1902) Developed the algebra of relations, still used today. 2 4 6

18. 16. Mathematical Linked to Relations Georg Cantor Father of set theory; introduced concepts of infinite sets and cardinality, establishing groundwork for understanding relations. René Descartes Invented the Cartesian coordinate system; the Cartesian product A × B forms the basis of defining relations. Bertrand Russell Advanced mathematical logic; formalized relations as fundamental mathematical objects. Augustus De Morgan Formalized relation theory through logic; De Morgan's laws describe relationships between sets and complements.

19. 17. Introduction to Relation Definition Let A and B be sets. A binary relation from A to B is a subset of A × B. Mathematically: R ⊆ A × B, where ( aRb ) denotes that (a, b) ∈ R, indicating that a is related to b by R. Example Let A={0,1,2} and B={a, b}. Then {(0,a),(0,b),(1,a),(2,b)} is a relation from A to B.

20. 18. Graphical Representation Symmetric Relation A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A. Antisymmetric Relation A relation R on a set A is such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a=b. Transitive Relation A relation R on a set A is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R for all a, b, c ∈ A. Empty Relation An empty relation on a set A contains no ordered pair at all.

21. Universal Relation A universal relation on a set A relates every element to every element of the set, including itself. Identity Relation An identity relation on a set A relates every element only to itself. Equivalence Relation A relation on a set A is an equivalence relation if it satisfies reflexivity, transitivity, and symmetry. Inverse Relation The inverse of a relation R from set A to set B is the set of all ordered pairs reversed from R.

22. 19. Properties of Relations 1 Empty and universal relations are symmetric and transitive. 2 Identity relations are reflexive, symmetric, and transitive (equivalence). 3 The intersection of two equivalence relations is also an equivalence relation.

23. 20. Mathematical Expressions Number of Relations If |A| = m and |B| = n, then the total number of relations from A to B is 2 ^ (mn). Example: If |A| = 2, |B| = 3, then relations = 2^6 = 64. Composition of Relations If R: A → B and S: B → C, then So R: A → C is defined as: (a, c) : ∃b ∈ B such that (a, b) ∈ R and (b, c) ∈ S.

24. 21. Summary of Relations A relation in Discrete Mathematics is a set of ordered pairs showing a connection between elements of two sets (a subset of A × B). It can be represented using arrow diagrams or tables. Main types include reflexive, symmetric, antisymmetric, and transitive relations. Some kinds are empty, universal, identity, equivalence, and inverse relations. An equivalence relation satisfies all three properties: reflexivity, symmetry, and transitivity. The total number of possible relations from A to B is 2^(m×n), where m and n are the

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