Set Symbols are a collective term used for all the symbols used in set theory which is the branch of mathematics that deals with the collection of objects and their various properties. A set is a well-defined collection of objects where each object in the collection is called an element and each element of the set follows a very specific rule. Generally, capital letter of English alphabets are used to denote sets and some letters denotes some specific sets in set theory.
There are many symbols used throughout the study of this branch of math, some of the common symbols are {}, |, :, ∈, ∉, ⊆, U, Ø, etc. We will discuss all these symbols in detail in the article including the history of these symbols as well. So, let’s start our journey of learning about various different set symbols used in set theory.
What are Set Symbols?
Set symbols are basic building blocks of mathematics that are used to represent and describe groups of objects, numbers, or items that have similar properties. These symbols offer a clear and consistent approach to communicating difficult ideas about sets and their interactions. The most typical set symbol is “∈,” which stands for “membership” and is pronounced as “belongs to”. “∈” indicates that an element is part of a specific set.
In contrast, “∉” signifies that an element does not form part of a set. ⊆, ⊂, ∪, ∩, ∅, etc. are some of the common examples of symbols in set theory. These and other symbols enable mathematicians to define operations, specify operations, and formulate exact mathematical assertions, laying the groundwork for a variety of mathematical specialities and practical uses.
Read more about Set Theory.
Example of Set Symbols
Let’s use the symbol, which stands for the intersection of sets, as an illustration. Let E and F be two sets such that Set E = {1, 3, 5, 7} and Set F = {3, 6, 9}. Then ∩ symbol represents the intersection between both sets i.e., E ∩ F.
Here, E ∩ F contains all the elements which are in common in both sets E and F i.e., {3}.
In conclusion, the ∩ symbol is used to identify the elements that are shared by two or more sets. The intersection only produces sets that have elements that are shared by all sets that are being intersected.
Learn more about Intersection of Sets.
History of Set Symbols
Between 1874 and 1897, a German mathematician called Georg Ferdinand Ludwig Philipp Cantor developed an abstract theory named Set Theory. He proposed it while researching some factual concerns involving specific forms of infinite sets of real numbers. A set, according to the notion, is a grouping of certain defined and distinct objects of observation. All of these things are referred to as members or components of the set. The property of real algebraic number combinations is the foundation of Cantor’s theory.
Basic Concepts of Set Symbols
Various ideas are covered at various levels of schooling in set theory. Set representation, set types, set operations (such as union and intersection), set cardinality and relations, and so on are among the essential concepts. Some of the essential concepts in set theory are as follows:
Universal Set
The capital letter ‘U’ is commonly used to represent a Universal Set. It is also occasionally symbolized by ε(epsilon). It is a set that contains all the elements of other sets as well as its own.
Complement of Set
The complement of a set comprises all of the universal set’s constituents except the elements of the set under examination. If A is a set, then its complements will contain all of the members of the specified universal set (U) that are not included in A. A set’s complement is indicated or expressed as A’ or Ac and is defined as:
A’= {x ∈ U: x ≠ A}
Read more about Complement of Set.
Set Builder Notation
Set Builder notation is the method to represent sets in such a way that where we don’t need to list all the elements of the set, we just need to specify the rule which is followed by all the elements of the set. Some examples of these notations are:
If A is a collection of real numbers.
A = {x : x ∈ R}
If A is a collection of natural numbers.
A = {x : x > 0 and x ∈ Z]
Where Z is set of Integers.
Read more, Representation of Sets.
Set Symbols in Maths
To refer to various things and amounts, the set symbol frequently uses a predefined list of variable symbols. To read and create set notation, you must first grasp how to employ symbols in diverse situations. Let us look at all the set theory notation and symbols relating to operations, relations, and so on, together with their meanings and examples, under this category.
Symbols Used in Number System
The symbols used in number systems are included in the table below:
| Whole Numbers |
These are the natural numbers. |
We know N = {1, 2, 3, . . . }
1 ∈ N
|
| Natural Numbers |
Natural numbers are sometimes referred to as counting numbers that begin with 1. |
We know W = {1, 2, 3, 4, 5, . . . }
0 ∈ W
|
| Integers |
Integers are comparable to whole numbers, except that they also include negative values. |
We know Z = {. . . , -3, -2, -1, 0, 1, 2, 3 . . .}
-6 ∈ Z
|
| Rational Numbers |
Rational numbers are those that are stated as a/b. In this case, a and b are integers with b ≠ 0. |
Q= {x | x=a/b, a, b ∈ Z and b ≠ 0}
2/6 ∈ Q
|
|
Irrational Numbers
|
Those number which can’t be represented in the form of a/b, are called irrational number i.e., all real number which are not rational.
|
P = {x | x ∉ Q}
π, e ∈ P
|
| Real Numbers |
Whole numbers, rational numbers, and irrational numbers make up real numbers. |
R= {x | -∞ < x <∞}
6.343434 ∈ R
|
| Complex Numbers |
A complex number is a combination of a real number and an imaginary number. |
C= {z | z = a + bi, a, b ∈ R}
6 + 2i ∈ C |
Set Theory Symbols
Delimiters are special characters or sequences of characters that indicate the beginning or end of a certain statement or function body of a specified set. The following are the delimiters set theory symbols and meanings:
| Set |
Within these brackets is a bunch of elements/ numbers/ alphabets in a set. |
{15, 22, c, d} |
| Such that |
These are used to construct a set by specifying what is contained within it. |
{ q | q > 6}
The statement specifies the collection of all q’s such that q is bigger than 6.
|
| Such that |
The “:” symbol is sometimes used instead of the “|” symbol. |
The above sentence can alternatively be written as {q | q > 6}. |
Sets and Relational Symbols in Sets Theory
Set theory symbols are used to identify a specific set as well as to determine/show a relationship between distinct sets or relationships inside a set, such as the relationship between a set and its constituent. The table below depicts such relationship symbols, along with their meanings and examples:
| Is a Component of |
This specifies that an element is a member of a specific set. |
If a set A={12, 17, 18, 27} we may say that 27 ∈ a. |
| Is not a Component of |
This indicates that an element does not belong to a particular set. |
If a set B={c, d, g, h, 32, 54, 59} then any element other than the one in the set does not belong to this set. As an example, 18 ∉ B. |
| Equality Relation |
The provided sets are equivalent in the sense that they have the same components. |
If you put P={16, 22, a} and Q={16, 22, a} then P=Q. |
| Subset |
When all of the items of A are present in B, A is a subset of B. |
A= {31, b} and B={a, b, 31, 54}
{31, b} ⊆ {a, b, 31, 54}
|
| Proper Subset |
P is said to be a proper subset of B when it is a subset of B and not equal to B. |
A= {24, c} and B={a, c, 24, 50}
A ⊂ B
|
| Not a Subset |
As a result, set A is not a subset of set B. |
A = {67,52} and B = {42,34,12}
A ⊄ B
|
| Superset |
A is a superset of B if set B is a subset of A. Set A can be the same as or greater than Set B. |
A = {14, 18, 26} and B={14, 18, 26}
{14, 18, 26} ⊇{14, 18, 26}
|
| Proper Superset |
Set A has more elements than set B since it is a superset of B. |
{14, 18, 26, 42} ⊃ {18,26} |
| Not a Superset |
When all of the elements of B are not present in A, A is not a true superset of B. |
A = {11, 12, 16} and B ={11, 19}
{11, 12, 16} ⊅ {11, 19}
|
| Empty Set |
An empty or null set is one that does not include any elements. |
{22, y} ∩ {33, a} = Ø |
| Universal Set |
A set that contains elements from all relevant sets, including its own. |
If, A = {a,b,c} and B = {1,2,3,b,c}, then U = {1,2,3,a,b,c} |
| Cardinality of a Set |
Cardinality refers to the number of items in a particular collection. |
If A= {17, 31, 45, 59, 62}, then |A|=5. |
| Power Set |
A power set is the set of all subsets of set X, including the set itself and the null set. |
If, X = {12, 16, 19}
P(X) = {12, 16, 19}={{}, {12}, {16}, {19}, {12, 16}, {16, 19}, {12, 19}, {12, 16, 19}}
|
Operator-based Symbols in Sets Theory
With examples, we will study set theory symbols and meanings for numerous operations such as union, complement, intersection, difference, and others.
| Union of Sets |
The union of sets creates an entirely new set by combining all of the components in the provided sets. |
A = {p, q, u, v, w}
B = {r, s, x, y}
A ∪ B (A union B) = {p, q, u, v, w, r, s, x, y}
|
| Intersection of Sets |
The common component of both sets is included in the intersection. |
A = { 4, 8, a, b} and B = {3, 8, c, b}, then
A ∩ B = {8, b}
|
| Complement of a set |
A set’s complement comprises all things that do not belong to the provided set. |
If A is universal set and A = {3, 6, 8, 13, 15, 17, 18, 19, 22, 24} and B = {13, 15, 17, 18, 19} then
X′ = A – B
⇒ X′ = {3, 6, 8, 22, 24}
|
| Set Difference |
The difference set is a set that contains items from one set that are not found in another. |
A = {12, 13, 15, 19} and B = {13, 14, 15, 16, 17}
A – B = {12, 19}
|
| Cartesian Product of Sets |
A Cartesian product is the product of the ordered components of the sets. |
A = {4, 5, 6} and B = {r}
Now, A × B ={(4, r), (2, r), (6, r)}
|
| Symmetric Difference of Sets |
A Δ B = (A – B) U (B – A) denotes the symmetric difference. |
A = {13, 19, 25, 28, 37},B = {13, 25, 55, 31}
A ∆ B = { 19, 28, 37, 55, 31}
|
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Solved Examples on Set Symbols
Example 1: Given two sets with P={21, 32, 43, 54, 65, 75} and Q={21, 43, 65, 75, 87, 98} what is the value of P∪Q?
Answer:
P={21, 32, 43, 54, 65, 75} and Q={21, 43, 65, 75, 87, 98}
P∪Q={21, 32, 43, 54, 65, 75, 87, 98}
Example 2: What is the value of |Y| if Y={13, 19, 25, 31, 42, 65}?
Answer:
|Y| = Cardinality of the set=number of elements in the set is the solution.
|Y| = n(Y)=6, since the set Y has 6 elements.
Example 3: Given two sets with values P={a,c,e} and Q={4,3}, determine their Cartesian product.
Answer:
Cartesian product = P × Q
If P={b, d, f} and Q={5, 6}
Then P × Q={(b,5), (d,6), (b,5), (d,6), (b,5), (d,6), (b,5), (d,6), (b,5), (d,6)}
Example 4: Assume P = {x: x is a natural integer and a multiple of 24, and Q = {x: x is a natural number smaller than 8}. Determine P ∪ Q.
Answer:
Given that
P = {1, 2, 3, 4, 6, 8, 12, 24}
Q = {1, 2, 3, 4, 5, 6, 7}
As a result, P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 12, 24}
Example 5: Assume P = {3, 5, 7}, Q = {2, 3, 4, 6}. Find (P ∩ Q)’.
Answer:
Given, P = {4, 6, 8}, Q = {3, 4, 5, 7}
P ∩ Q = {4}
Therefore,
(P ∩ Q)’ = {3, 5, 6, 7, 8}
Example 6: If P = {4, 5, 7, 8, 9, 10} and Q = {3, 5, 7, 9, 12, 14}, determine
(i) P-Q and (ii) P-Q.
Answer:
Given,
P = {4, 5, 7, 8, 9, 10} and Q = {3, 5, 7, 9, 12, 14}
(i) P – Q = {4, 8, 10}
(ii) Q – P = {3, 12, 14}
Practise Questions for Set Symbols
Question 1: Given the sets:
- A = {2, 4, 6, 8}
- B = {4, 8, 12, 16}
Determine the elements in the union of sets A and B.
Question 2: Let’s consider the sets:
- X = {1, 2, 3, 4, 5}
- Y = {3, 4, 5, 6, 7}
Find the intersection of sets X and Y.
Question 3: Suppose you have the sets:
- P = {a, b, c, d}
- Q = {c, d, e, f}
Calculate the elements in set P – Q as well as Q – P.
Question 4: Let’s say you have the sets:
- U = {1, 2, 3, 4, 5}
- V = {4, 5, 6, 7}
Find out whether set V is a subset of set U.
Question 5: Consider the sets:
- S = {apple, banana, orange, pear}
- T = {pear, mango, cherry}
Find the Cartesian product of sets S and T.
Question 6: Suppose you have the universal set:
- U = {a, b, c, d, e, f, g, h, i, j}
And the sets:
- E = {b, d, f, h, j}
- F = {a, c, e, g, i}
Calculate the complement of set E and F with respect to the universal set U.
FAQs on Set Symbols
1. Define Set Symbol.
The set symbol is a branch that studies groupings of entities/numbers/objects, their relationships with other sets, different operations (union, intersection, complement, and difference), and associated features.
2. What does this symbol ⊆ represent?
The symbol “⊆” means “is a subset of.” A subset is a set whose items have been added as if they were all elements of another set.
3. What does ∪ mean in sets?
‘∪’ is the sign for the set union. A ∪ B is a set that contains all of the elements of sets A and B.
4. What does P = Q represents?
If set P equals set Q, then the members of P and Q are the same. For example:
P = {4,5,6} and Q = {6,5,4}
As a result, P = Q.
5. In mathematics, what does ∩ mean?
‘∩’ signifies the union of two sets. A ∩ B is a set that contains items shared by both A and B.
6. What is ∈ in sets?
∈ is a sign that means ‘belongs to’. If b ∈ B, it indicates that b is an element of B.
7. What is the set N ={1, 2, 3, 4, 5, . . .} known as?
The set of natural numbers is defined as N = {1, 2, 3, 4, 5, …} It contains all positive numbers, ranging from 1 to an infinite number. This collection is crucial to mathematics and provides a framework for both ordering and counting.
8. What is A × B in sets?
The Cartesian product of the sets A and B is shown as A x B in the set symbol. It is the set that includes all possible ordered pairings in which the first element is drawn from set A and the second from set B.
9. How will you read A ∩ B?
A∩B is pronounced “A intersection B.” It stands for the set that contains elements which are common in both sets.
10. What does the Ø mean in Set Theory?
In set theory, the idea of an empty set, which has no items, is denoted by the symbol Ø(pronounced “empty set”).
11. What is AUB?
AUB in mathematics stands for the union of the sets A and B. It refers to the set that includes every element from both sets A and B.
12. Is ∅ the same as {}?
Yes, ∅ and {} both represent the empty set in mathematics. Thus, both are the different notation of same thing.
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