Set symbols are special mathematical symbols used in set theory. These symbols help to describe relationships between sets, elements, and operations involving sets.
Some of the commonly used set symbols:
The symbols used in number systems are included in the table below:
| Symbol | Name | Meaning/Definition | Example |
|---|---|---|---|
| W | Whole Numbers | Non-negative integers including zero. | We know W = {1, 2, 3, 4, 5, . . . } 0 ∈ W |
| N or ℕ | Natural Numbers | Natural numbers are sometimes referred to as counting numbers that begin with 1. | We know N = {1, 2, 3, . . . } 1 ∈ N |
| Z or ℤ | 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 |
| Q or ℚ | 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 |
P or ℙ | 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 | |
| R or ℝ | Real Numbers | Whole numbers, rational numbers, and irrational numbers make up real numbers. | R = {x | -∞ < x <∞} 6.343434 ∈ R |
| C or ℂ | 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 |
Basic Set Notation
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:
| Symbol | Name | Meaning/Definition | Example |
|---|---|---|---|
| {} | 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}. |
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:
| Symbol | Name | Meaning/Definition | Example |
|---|---|---|---|
| a ∈ A | 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 |
| b ∉ B | 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 |
| A = 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 |
| A ⊆ B | 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} |
| A ⊂ B | 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 |
| 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 |
| 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} |
| A ⊃ B | Proper Superset | Set A has more elements than set B since it is a superset of B. | {14, 18, 26, 42} ⊃ {18,26} |
| A ⊅ B | 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} |
| Ø | Empty Set | An empty or null set is one that does not include any elements. | {22, y} ∩ {33, a} = Ø |
| U | 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} |
| |A| or n{A} | 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 |
| P(X) | 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}} |
Set Operations Symbols
With examples, we will study set theory symbols and meanings for numerous operations such as union, complement, intersection, difference, and others.
| Symbol | Name | Meaning/Definition | Example |
|---|---|---|---|
| A ∪ B | 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} |
| A ∩ B | 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} |
| Xc OR X' | 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} |
| A − B | 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} |
| A × B | Cartesian Product of Sets | A Cartesian product is the product of the ordered components of the sets. | A = {4, 5, 6} and B = {r} A × B = {(4, r), (5, r), (6, r)} |
| A ∆ B | 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} |