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In Set Theory, we use elements inside sets as discrete elements. We can consider them as discrete elements in discrete mathematics. In several use-cases of discrete mathematics, we use sets. Sets are the foundational building blocks in discrete mathematics.
In this chapter, we will have a detailed look at the concept of sets, their notations, and the various operations that can be performed on them.
Definition of a Set
A set is an unordered collection of objects. These objects are called elements or members of the set. They can be anything: numbers, letters, other sets, or even abstract concepts.
In a set, order does not matter and repetition is ignored (an element is either in the set or not). Following are some examples of sets –
- The set of all actors who have played The Doctor on Doctor Who
- The set of natural numbers between 1 and 10 inclusive
Set Equality
Another important idea of sets is set equality. Two sets are considered equal if and only if they contain exactly the same elements. For instance, the set of vowels in the word "questionably" is equal to the set of vowels (Since it has all of the 5 vowels, {a, e, i, o, u}).
Set Notation
Here we will see the different types of set notations used in mathematics with their use‑cases and benefits.
| Set Notation | Example and Descriptions |
|---|---|
| Element Notation | We use the symbol ∈ to denote that an object is an element of a set. Example − a ∈ {a, b, c} It is read as "a is an element of the set containing a, b, and c." Conversely, we use ∉ to denote that an object is not an element of a set. Example − d ∉ {a, b, c} |
| Set Builder Notation | Set builder notation is used to describe sets, especially when listing all elements is impractical. The general form is: {x : condition(s) that x must satisfy} Example − A = {x ∈ N : ∃n ∈ N(x = 2n)} This is read as "A is the set of all x in the natural numbers such that there exists some n in the natural numbers for which x is twice n." In simpler terms, A is the set of all even natural numbers. A simpler way to write this could be − A = {x ∈ N : x is even} |
Special Sets
There are some special types of sets that play an important role in set theory −
| Special Sets | Descriptions |
|---|---|
| Empty Set | The empty set, denoted by ∅, is the set containing no elements. |
| Universal Set | The universal set, often denoted by U, is the set of all elements under consideration. |
| Common Number Sets |
|
Subsets
A subset can either be an ordinary subset or a proper subset.
| Subset Types | Descriptions |
|---|---|
| Subset | A set A is a subset of set B (written A ⊆ B) if every element of A is also an element of B. |
| Proper Subset | A set A is a proper subset of set B (written A ⊂ B) if A ⊆ B and A ≠ B. |
Set Operations
The operations on sets are given in the following table in short. We will see the operations with examples in a later chapter in detail.
| Operations | Descriptions |
|---|---|
| Intersection | The intersection of sets A and B (written A ∩ B) is the set containing all elements that are in both A and B. |
| Union | The union of sets A and B (written A ∪ B) is the set containing all elements that are in A or B or both. |
| Cartesian Product | The Cartesian product of sets A and B (written A × B) is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. |
| Set Difference | The set difference of A and B (written A \ B) is the set containing all elements of A that are not in B. |
| Complement | The complement of set A (written A̅) is the set of all elements in the universal set that are not in A. |
Some other concepts in sets are given below −
| Operations | Descriptions |
|---|---|
| Power Set | The power set of A (written P(A)) is the set of all subsets of A. |
| Cardinality | The cardinality of a set A (written |A|) is the number of elements in A. |
Example of Related to Sets
Let us take a look at some of the examples of these topics related to sets for a better understanding −
- Describe the set {x : x + 3 ∈ N} − This is the set of all numbers which, when 3 is added to them, result in a natural number. The set could be written as {-3, -2, -1, 0, 1, 2, ...}.
- Describe the set {x ∈ N : x + 3 ∈ N}: Solution − This is the set of all natural numbers which, when 3 is added to them, result in a natural number. So here we just have {0, 1, 2, 3 ...}.
- Describe the set {x : x ∈ N ∨ -x ∈ N} − This is the set of all integers (positive and negative whole numbers, written Z). In other words, {... , -2, -1, 0, 1, 2, ...}.
- Describe the set {x : x ∈ N ∧ -x ∈ N} − Here we want all numbers x such that x and -x are natural numbers. There is only one: 0. So we have the set {0}.
Set Theory Notation Table
All the symbols that are used in sets are given in the following table, for a quick reference −
| Symbol | Meaning |
|---|---|
| {, } | Used to enclose elements of a set |
| : | Such that |
| ∈ | Is an element of |
| ∉ | Is not an element of |
| ⊆ | Is a subset of |
| ⊂ | Is a proper subset of |
| ∩ | Intersection |
| ∪ | Union |
| × | Cartesian product |
| \ | Set difference |
| A̅ | Complement of A |
| |A| | Cardinality of A |
| ∅ | Empty set |
| U | Universal set |
| N | Set of natural numbers |
| Z | Set of integers |
| Q | Set of rational numbers |
| R | Set of real numbers |
| P(A) | Power set of A |
Conclusion
In this chapter, we explained the characteristics of sets and understood how to represent them using various notations, including element notation and set builder notation. We highlighted the special sets like the empty set and universal set as well as common number sets.
In addition, we provided an overview of important set operations and relations such as subsets, intersections, unions, and complements. Through examples, we explained how to interpret and describe sets in different forms.
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