Set theory is an important part of mathematics, providing a comprehensive and powerful tool for well-organizing groups of objects. It is about the idea and practice in which we look at groups of objects as “sets” and study the relationships between them. Set theory is used in mathematics, science, higher education, and many other fields. In this article, we will focus on the basic concepts, uses, and applications of set theory.
set theory
Set theory is a branch of logical mathematics that studies collections of objects and operations based on them.
A set is simply a collection of objects or a group of objects. For example, a group of players on a football team is a set and the players on the team are its objects.
definition of set
A set is defined as “a well-defined collection of objects”. Suppose we have a set of natural numbers, then all natural numbers will be its members and the collection of numbers is well defined in that they are natural numbers. A set is always represented with a capital letter. We use curly brackets {} to represent sets and commas (,) to separate things in a set.
For example A = {1, 2, 3, 4, a, b, c, d} is a set in which 1, 2, 3, 4, a, b, c, d are the elements of this set.
examples of aggregates
Some common examples of aggregates are given below:
- Set of natural numbers: N = {1, 2, 3, 4…}
- Set of even numbers: E = {2, 4, 6, 8…}
- Set of prime numbers: P = {2, 3, 5, 7,….}
- Set of integers: Z = {…, -4, -3, -2, -1, 0, 1, 2,….}
representation of sets
Sets are mainly represented in two forms:
- roster form
- set builder form
roster form
In the roster form of a set, the elements are placed inside {} and separated by commas. Suppose we have to represent a set of natural numbers in roster form then its roster form is given as N = {1, 2, 3, 4…..}.
In roster form representation, the set does not contain the same elements. For example, if A represents a set that contains all the letters of the word TREE, the correct roster form set would be:
A = {T,R,E}
= {E,R,T}
A = {T,R,E,E} is a wrong representation,
Therefore A ≠ {T,R,E,E}
set builder form
In set builder form, instead of writing the elements directly inside the curly brackets, a rule or statement describing the common characteristics of all elements is written. For example,
- A set of all prime numbers less than or equal to 10
P = {p : p is a prime number ≤ 10} - set of natural numbers
N = {n : n is a natural number}
types of aggregates
The sets are further classified into different types depending on the elements or types of elements. These are the different types of sets in basic set theory:
| types of aggregates | ||
| types of aggregates | Description | Example |
|---|---|---|
| empty set | A set that does not contain any elements. | , |
| single set | A set that contains a single element. | {1} |
| finite set | A set with finite, countable elements | {apple, banana, orange} |
| infinite set | set with infinite number of elements | {Natural numbers (1, 2, 3, …)} |
| equivalent set | Sets which have the same number of elements and their elements can be related one-to-one. | Set A = {1, 2, 3} and set B = {a, b, c} (assuming that a is related to 1, b to 2, and c to 3.) |
| similar set | Sets that have the same elements. | Set A = {1, 2} and set B = {1, 2} |
| universal set | A set that includes all the elements relevant to a specific discussion. | The group of all students in a school (when student grades are discussed) |
| unequal set | Sets that do not all have the same elements. | Set A = {1, 2, 3} and set B = {a, b} |
| power set | A set consists of all possible subsets of a given set. | Power set of {a, b} = { {}, {a}, {b}, {a, b} } |
| Subset | A set where all the elements are also members of another set. | {1, 2}, a subset of {1, 2, 3} |
symbols of set theory
Various symbols are used in set theory. The notations and their explanations are tabulated below:
| symbols of set theory | |
| Sign | the explanation |
|---|---|
| , | set of |
| x∈A | x is an element of the set A |
| x ∉ A | x is not an element of the set A |
| ∃ or ∄ | it exists or does not exist |
| Φ | empty set |
| A = B | similar set |
| n(A) | Cardinal numbers of set A |
| P(A) | power set |
| A ⊆ B | A is a subset of B |
| A ⊂ B | A is a proper subset of B |
| A ⊈ B | A is not a subset of B |
| B ⊇ A | B is a superset of A |
| B ⊃ A | B is a proper superset of A |
| B ⊉ A | B is not a superset of A |
| A ∪ B | A Insertion B |
| A ∩ B | A universal B |
| A' | Complement of set A |
operations on sets
There are four important widely used set operations:
| operations on sets | ||
| set operations | Description | Example |
|---|---|---|
| Union (U) | Combination of elements from two sets including common elements | If set A = {1, 3, 5} and set B = {2, 3, 4}, then AUB = {1, 2, 3, 4, 5} |
| Intersection (∩) | set of elements in both sets | If set A = {1, 3, 5} and set B = {2, 3, 4}, then A ∩ B = {3} |
| Difference ( \ ) | The set of elements of the first set that are not in the second set. | If set A = {1, 3, 5} and set B = {2, 3, 4}, then A \ B = {1, 5} |
| Complement (A') | The collection of elements that are not in the first set, but are in the universal set. | If universal set = {1, 2, 3, 4, 5} and set A = {2, 4}, then A' = {1, 3, 5} |
| Cartesian Multiplication (A x B) | A collection of ordered pairs where the first element comes from the first set and the second element comes from the second set. | If set A = {1, 2} and set B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b) |
set theory formulas
| set theory formulas |
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Properties of set operations
The various properties followed by sets are tabulated below:
| Properties of set operations | |
| Property | expression |
|---|---|
| commutative properties | A ∪ B = B ∪ A A ∩ B = B ∩ A |
| associative properties | (A ∩ B) ∩ C = A ∩ (B ∩ C) (A ∪ B) ∪ C = A ∪ (B ∪ C) |
| distributive properties | A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| identity properties | A ∪ Φ = A A ∩ U = A |
| complementary properties | A ∪ A' = U |
de morgan's laws
De Morgan's law applies in relating the union and intersection of two sets through their complements. There are two rules under De Morgan's law.
- De Morgan's law of union – De Morgan's law of union states that the complement of the union of two sets is equal to the intersection of the complements of the individual sets. This can be expressed mathematically as follows.
(A ∪ B)' = A' ∩ B'
- De Morgan's law of intersection – De Morgan's intersection rule states that the complement of the intersection of two sets is equal to the union of the complements of the individual sets. Mathematically it can be expressed as follows.
(A ∩ B)' = A' ∪ B'
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