MAT 111 Foundations of Mathematics

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Browsing MAT 111 Foundations of Mathematics by Author "Michael Obiero"

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    MAT 111 Module 1: Sets
    (2023-04-22) Michael Obiero
    This module introduces the concepts of sets, which are the building blocks of almost all mathematical concepts and statements. Once we have a well defined set, we can construct relations on elements of that set, or between sets, and functions between sets. In this module you will be introduced to operations between sets, cardinality of sets and classification of sets.
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    MAT 111 Module 2: Relations and functions
    (2023-04-22) Michael Obiero
    This module introduces the concepts of relations on sets and between sets. You will be presented with different types of relations. A very important relation is the equivalence relations which is related to partitions of sets. Another important relation is a `function' which is a special relation between elements of two sets that maps all elements of the domain to unique elements in the range. Functions are important in many ways and here, you will be introduced to the basics of what a function is and some properties and examples of functions that are common in mathematics.
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    MAT 111 Module 3: Trigonometry
    (2023-04-22) Michael Obiero
    In the previous lesson on "Relations and functions" we defined and looked at examples of functions. Some of the most important functions with wide applications in engineering, computing, finance e.t.c. are trigonometric functions. Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums, motion of objects, population dynamics. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this lesson,you will be introduced to trigonometric functions like \(\sin(x)\), \(\cos(x)\), \(\tan(x)\) from the right angled triangle their use in calculating angles and lengths. You will then be introduced to trigonometric identities and their applications in solving trigonometric equations. Lastly, you will learn about graphs and periods of trigonometric functions, and shifted graphs of sine and cosine functions.
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    MAT 111 Module 4: Polynomials
    (2023-04-22) Michael Obiero
    In this module you will be introduced to polynomials, different types and properties of polynomials depending on their degree, factorization of polynomial functions, techniques of finding roots of polynomial functions, factorization of polynomials of degree 2 also called the quadratic functions. You also study methods of solving quadratic equations and inequalities.
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    MAT 111 Module 5: Complex Numbers
    (2023-04-22) Michael Obiero
    In the previous module on "Polynomials" we studied roots of polynomial equations and how to calculate for the roots of polynomials. There are however instances where solving equations like \(x^{2}+1=0\) does not yield any real solutions. This is one of the main motivations of complex numbers. In this module, you will learn about complex numbers and their algebraic properties of addition, multiplication and division. You will also learn about cartesian and polar representation of complex numbers, which leads to evaluation of powers and roots of complex numbers. Some further applications of complex numbers will be presented.
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    MAT 111 Module 6: Vectors and Matrices
    (2023-04-22) Michael Obiero
    In the previous module on "Complex Numbers" you learned that a complex number can be represented as a vector. That is, a quantity with both magnitude and direction. Vectors play a very important role, and matrices are an example of functions between sets of vectors. In this module, we will formally introduce the concept of vectors, operations on vectors solve problems involving vectors. It will be important to write these vectors in matrix form to make calculations easier. In most aspects of life, you will encounter linear equations. In most applications, these equations can be rather large and will need to be converted to matrices so that they can be solved. You will be introduced to matrices, albeit smaller ones, and learn basic numerical and algebraic operations on matrices. Some of the most interesting applications of matrices will be learned at a later stage, but here you will be introduced to basic operations. The most important, and widely applied property of matrices that you will learn in this module is the determinant of a square matrix.
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    MAT 111 Module 7: Lines and area of geometric shapes
    (2023-04-22) Michael Obiero
    This module introduces concepts of straight lines in the plane, the distance between a point and a line, parallel and perpendicular lines and angle measurements. You will learn about regular polygons and area calculations involving triangles, rectangles, trapeziums and trapezoids among other polygons. These geometric objects have many applications in our daily lives from construction, carpentry, engineering where length and angle measurements are critical.
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    MAT 111 Module 8: Conics Sections
    (2023-04-22) Michael Obiero
    In this module, you will learn about the two-dimensional figures that are formed when a right circular cone is intersected by a plane. These include the ellipse, the parabola and the hyperbola. You will learn how to develop defining equations for each figure and then learn how to use these equations to solve a variety of problems.
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    MAT 111 Module 9: Counting: Permutations and Combinations
    (2023-04-22) Michael Obiero
    This module introduces develops techniques of determining, without direct enumeration, the number of elements in a set, or the number of possible outcomes of an event. Such counting includes the study of permutations and combinations. In this module, you will be introduced to the following concepts: \begin{itemize} \item Basic counting principles, \item The binomial theorem and Pascal's triangle, \item Permutations and combinations, \item The Pigeonhole principle and its applications. \end{itemize}