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 ↓


Sets and Subsets


Before we dive into the world of sets and subsets within set theory and logic, let's lay the foundation for understanding what a set is. In mathematics, a set is simply a collection of distinct objects, treated as a unit. These objects can be anything, numbers, symbols, or even other sets. The main property that defines a set is that it must be well-defined. This means that you should be able to clearly define what the set consists of. For example, the collection of planets in our solar system forms a set.

Basic notation and terminology for sets

When representing sets, it is common to use curly braces { }. The objects inside the set, known as elements or members, are listed inside these braces. For example, if we have a set A containing the numbers 1, 2, and 3, it is represented like this:

A = {1, 2, 3}

Sets can also be defined using descriptive language inside braces. If the set B contains all even numbers less than 10, we write it as:

B = {2, 4, 6, 8}

Alternatively, we can use shorthand by specifying a property to be satisfied by the members of the set. For example:

C = {x | x is an even number less than 10}

The vertical bar | means "such that". Let's illustrate sets with a simple example:

A123

In this visual example, the circle represents a set called A, which contains the elements 1, 2, and 3.

Defining subsets

A subset is a part of a set that contains some or all of its elements. If all the elements of a set A are also in a set B, then A is a subset of B We represent this as follows:

A ⊆ B

If A is a subset of B, but A is not equal to B (that is, B contains elements that are not in A), then A is a proper subset of B:

A ⊂ B

To illustrate this concept with a more concrete example:

Suppose the set X = {a, b, c}, and the set Y = {a, b, c, d, e}. It is clear that every element of the set X is also in the set Y Therefore, X is a subset of Y, which is written as:

X ⊆ Y

However, since the set X does not contain all the elements of the set Y (in particular, 'd' and 'e' are missing), X is a proper subset of Y:

X ⊂ Y

Let's imagine a subgroup:

ABCABCDE

In this diagram, the smaller circle represents the set X, which shows that it is completely within the larger circle representing the set Y This visualization again shows that X is a subset of Y

Types of sets

It's important to understand the different types of sets to set the stage for upcoming discussions. Here are the types you may encounter:

  • Empty set: Also known as the null set, it has no elements. It is represented by ∅ or {}. The empty set is a subset of every set.
  • Singleton set: A set with only one element, for example, {a}.
  • Finite set: A set with a finite number of elements, such as {1, 2, 3}.
  • Infinite set: A set with an infinite number of elements, such as the set of all natural numbers {1, 2, 3, ...}.
  • Universal set: In any given context, the universal set is the set of all possible elements. It generally includes other sets, and is denoted by U The contents of the universal set depend on the context. It can be the set of all integers, all real numbers, etc.

For example, consider these sets:

A = ∅ B = {5} C = {1, 2, 3, 4, 5} D = {1, 2, 3, ...} U = {all integers}

Each of these sets conforms to the description given above. Set A is the empty set. Set B is a singleton set. Set C is a finite set. Set D is an infinite set, and set U is the universal set.

Operations on sets

There are many operations that can be performed on sets, which are fundamental to understanding subsets and the relationships between sets. Here are some simple but important operations:

Union

The union of two sets is another set that contains all the elements of both sets. The symbol for union is ∪. If A and B are sets, then their union is denoted as A ∪ B For example, if A = {1, 2, 3} and B = {3, 4, 5}, then:

A ∪ B = {1, 2, 3, 4, 5}

Intersection

The intersection of two sets is the set of all elements that are common to both sets. The symbol for intersection is ∩. For sets A and B, the intersection is represented as A ∩ B If A = {1, 2, 3} and B = {3, 4, 5}:

A ∩ B = {3}

Difference

The difference of two sets A and B is the set of all elements that are in A but not in B It is denoted as A - B Using our previous sets A and B:

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

Complement

The complement of a set A is the set of all elements of the universal set U that are not in A It is often denoted as A' or A^c. Let U = {0, 1, 2, 3, 4, 5} and A = {1, 2, 3}:

A' = {0, 4, 5}

Power set

The power set of a set A is the set of all possible subsets of A, including A itself and the empty set. If A has n elements, then its power set will have 2^n elements.

Consider the set A = {1, 2}. The power set of A is:

ℙ(A) = {{}, {1}, {2}, {1, 2}}

Notice all four possible subsets, including the empty set {} and the set A itself.

Logical set relations

Set theory overlaps a lot with logic, and understanding their interrelationship can be quite insightful. Relations between sets express logical relationships, which can be important in various proofs and logical assessments.

Equality of sets

Two sets, A and B, are said to be equal if their elements are exactly the same. This means:

A = B ⇔ (∀x(x ∈ A ⇔ x ∈ B))

In other words, if every element of A is in B and every element of B is in A, then the two sets are equal.

Subgroup creation

One of the logical relations is the formation of subsets. A set A is a subset of a set B if for every element x in A, x is also in B

Mathematically it is expressed as:

A ⊆ B ⇔ (∀x(x ∈ A ⇒ x ∈ B))

Real-World Applications of Sets and Subsets

Understanding sets and subsets is not just an academic exercise. They play a vital role in various fields including computer science, probability theory, statistics, and others. For example, in database management, we often deal with sets of data. Understanding how to manipulate these sets through union, intersection, and difference operations is crucial for data queries and data set operations.

Similarly, in logic gates and circuits, sets are used to establish connections between inputs and outputs, representing operations such as AND, OR, and NOT in data processing.

By mastering the concepts of sets and subsets, you equip yourself with the mathematical tools needed to understand these and other complex systems in both theoretical and practical situations.

Conclusion

The study of sets and subsets is foundational to further exploration in mathematics and its applications. With an understanding of how to define and manipulate sets, one can better engage with higher-level mathematical concepts such as functions, relations, and algebraic structures. The relevance of set theory is wide-ranging, touching computer science, data management, logic, and many other fields, illustrating the interconnectedness of mathematical ideas. Developing proficiency with sets and subsets can thus enhance one's problem-solving arsenal in both pure and applied mathematics.


Undergraduate → 13.1


U
username
0%
completed in Undergraduate

← Prev (13)
Set Theory and Logic
Next (13.2) →
Relations and Functions

Comments 



Sets and Subsets

https://www.buddymath.com/ug/13.1?bmal=en