Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Set Theory and Logic ↓


Relations and Functions


In undergraduate mathematics, set theory and logic form the basis for understanding various mathematical concepts, including relations and functions, which are crucial for understanding complex structures and operations. This document provides a comprehensive guide to these concepts with many examples.

Introduction to sets

A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. A set can contain numbers, letters, symbols, or even other sets. For example:

A = {1, 2, 3, 4}

Here, A is a set containing the elements 1, 2, 3, and 4. The order of the elements does not matter, and sets do not contain duplicate elements.

Sets are written with curly braces. If the set is empty, it is called the empty set, and is represented by {} or ∅.

Relationship

A relation is an association between the elements of two sets. It is usually represented as a subset of the Cartesian product of these sets. If there are two sets, A and B, then the Cartesian product A × B is the set of all possible ordered pairs (a, b), where a ∈ A and b ∈ B.

For example, let us consider the following sets:

A = {1, 2, 3} B = {4, 5}

The Cartesian product A × B is:

A × B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

A relation R from a set A to a set B is a subset of A × B. For example, let us define a relation R such that an element of A is related to an element of B if the sum of the two elements is greater than 5:

R = {(1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

Visually, we can represent relationships using a type of graph called a directed graph or arrow diagram.

1 2 3 4 5

Properties of relations

Relations may have certain special properties such as reflexivity, symmetry, transitivity, etc., especially when the relation is defined on a set together with itself (a relation from A to A).

  • Reflexive: A relation R on a set A is reflexive if every element belongs to itself. Formally, (a, a) ∈ R for all a ∈ A
  • Symmetric: A relation R on a set A is symmetric if (a, b) ∈ R implies that (b, a) ∈ R for all a, b ∈ A
  • Transitive: A relation R on a set A is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

Work

A function is a specific type of relationship between two sets A and B, where each element of set A is related to exactly one element of set B. Functions can be represented using expressions, graphs, and ordered pairs.

Symbolically, a function f from a set A to a set B is represented as f: A → B We say that f(x) is the image of x under f.

f: A → B f(x) = x^2 A = {1, 2, 3} B = {1, 4, 9}

In the above example, we see that each element of A is mapped to an element of B, specifically its class.

1 2 1 4 9

Types of tasks

  • One-to-one (injective) function: A function is injective if distinct elements in the domain correspond to distinct elements in the range. Formally, f(a) = f(b) implies a = b.
  • Inner (surveyor) function: A function is surveyor if every element of the codomain is the image of at least one element of the domain.
  • Bijective: A function is bijective if it is both injective and surjective, meaning that it establishes a perfect "pairing" between the domain and codomain.

Formal definitions and examples

A formal definition provides the rigor and basis on which real-world applications can be evaluated.

One-to-one (injective) example

Consider the function f: R → R defined by f(x) = 2x + 3 Let us prove that this function is injective.

Suppose f(x1) = f(x2). Then, 2x1 + 3 = 2x2 + 3. Subtract 3 from both sides, 2x1 = 2x2. Divide by 2, x1 = x2.

Since we have proved x1 = x2, the function is injective.

Onto (surjective) example

Consider the function f: R → R defined by f(x) = x^3. We need to show that the function is omniprojective.

Let y be a real number such that y = x^3. Then, x = y^(1/3) satisfies this.

Thus, for any y in the codomain, there exists an x in the domain such that f(x) = y. This makes the function omniscient.

Bifurcation example

Consider the function f: R → R defined by f(x) = x^2. Let us determine whether it is bijective.

f(x1) = f(x2), implies x1^2 = x2^2, thus x1 = x2 or x1 = -x2. Not one-to-one (injective is false). The range of f(x) = x^2 is only non-negative real numbers. Hence it's not onto (surjective is false).

Therefore, f(x) = x^2 is not a double bond.

Important concepts in works

  • Domain: The set of all input values (elements from the set A) for which the function is defined.
  • Codomain: the set of all possible output values (not all may be mapped from the domain by the function).
  • Range: The set of all actual output values generated by the function from the input values in the domain.

Function composition and inverses

Function composition: The combination of two functions where the output of one function becomes the input of the other function. If two functions f: A → B and g: B → C, then the composition (g ∘ f): A → C is defined by (g ∘ f)(x) = g(f(x)).

Inverse function: If f: A → B is a bijection, then its inverse f -1 : B → A reverses the mapping and holds f(f -1 (x)) = x for all x ∈ B, and f -1 (f(y)) = y for all y ∈ A

Consider the example function:

Let f(x) = 2x - 3 and g(x) = x + 3/2. Result of (g ∘ f)(x) = g(f(x)). f(f -1 (x)) should map back original value.

With the above explanations and examples, we have tried to cover the essence of relations and functions in the context of set theory and mathematical logic. Understanding these concepts forms the basis for more advanced mathematical topics, models, and real-world applications.


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