Sets
| Sign | Example | Meaning and verbal equivalent | Remarks |
|---|---|---|---|
| ∈ | x ∈ A | x belongs to A; x is an element of the set A | |
| ∉ | x ∉ A | x does not belong to A; x is not an element of the set A | The negation stroke can also be vertical. |
| ∋ | A ∋ x | the set A contains x (as an element) | same meaning as x ∈ A |
| ∌ | A ∌ x | the set A does not contain x (as an element) | same meaning as x ∉ A |
| { } | {x1, x2, ..., xn} | set with elements x1, x2, ..., xn | also {xi ∣ i ∈ I}, where I denotes a set of indices |
| { ∣ } | {x ∈ A ∣ p(x)} | set of those elements of A for which the proposition p(x) is true | Example: {x ∈ ℝ ∣ x > 5} The ∈A can be dropped where this set is clear from the context. |
| card | card(A) | number of elements in A; cardinal of A | |
| ∖ | A \ B | difference between A and B; A minus B | The set of elements which belong to A but not to B. A ∖ B = { x ∣ x ∈ A ∧ x ∉ B } A − B should not be used. |
| ∅ | the empty set | ||
| ℕ | the set of natural numbers; the set of positive integers and zero | ℕ = {0, 1, 2, 3, ...} Exclusion of zero is denoted by an asterisk: ℕ* = {1, 2, 3, ...} ℕk = {0, 1, 2, 3, ..., k − 1} |
|
| ℤ | the set of integers | ℤ = {..., −3, −2, −1, 0, 1, 2, 3, ...} ℤ* = ℤ \ {0} = {..., −3, −2, −1, 1, 2, 3, ...} |
|
| ℚ | the set of rational numbers | ℚ* = ℚ \ {0} | |
| ℝ | the set of real numbers | ℝ* = ℝ \ {0} | |
| ℂ | the set of complex numbers | ℂ* = ℂ \ {0} | |
| closed interval in ℝ from a (included) to b (included) | = {x ∈ ℝ ∣ a ≤ x ≤ b} | ||
| ],] (,] |
]a,b] (a,b] |
left half-open interval in ℝ from a (excluded) to b (included) | ]a,b] = {x ∈ ℝ ∣ a < x ≤ b} |
| [,[ [,) |
[a,b[ [a,b) |
right half-open interval in ℝ from a (included) to b (excluded) | [a,b[ = {x ∈ ℝ ∣ a ≤ x < b} |
| ],[ (,) |
]a,b[ (a,b) |
open interval in ℝ from a (excluded) to b (excluded) | ]a,b[ = {x ∈ ℝ ∣ a < x < b} |
| ⊆ | B ⊆ A | B is included in A; B is a subset of A | Every element of B belongs to A. ⊂ is also used. |
| ⊂ | B ⊂ A | B is properly included in A; B is a proper subset of A | Every element of B belongs to A, but B is not equal to A. If ⊂ is used for "included", then ⊊ should be used for "properly included". |
| ⊈ | C ⊈ A | C is not included in A; C is not a subset of A | ⊄ is also used. |
| ⊇ | A ⊇ B | A includes B (as subset) | A contains every element of B. ⊃ is also used. B ⊆ A means the same as A ⊇ B. |
| ⊃ | A ⊃ B. | A includes B properly. | A contains every element of B, but A is not equal to B. If ⊃ is used for "includes", then ⊋ should be used for "includes properly". |
| ⊉ | A ⊉ C | A does not include C (as subset) | ⊅ is also used. A ⊉ C means the same as C ⊈ A. |
| ∪ | A ∪ B | union of A and B | The set of elements which belong to A or to B or to both A and B. A ∪ B = { x ∣ x ∈ A ∨ x ∈ B } |
| ⋃ | union of a collection of sets | , the set of elements belonging to at least one of the sets A1, …, An. and, are also used, where I denotes a set of indices. | |
| ∩ | A ∩ B | intersection of A and B | The set of elements which belong to both A and B. A ∩ B = { x ∣ x ∈ A ∧ x ∈ B } |
| ⋂ | intersection of a collection of sets | , the set of elements belonging to all sets A1, …, An. and, ⋂i∈I are also used, where I denotes a set of indices. | |
| ∁ | ∁AB | complement of subset B of A | The set of those elements of A which do not belong to the subset B. The symbol A is often omitted if the set A is clear from context. Also ∁AB = A \ B. |
| (,) | (a, b) | ordered pair a, b; couple a, b | (a, b) = (c, d) if and only if a = c and b = d. ⟨a, b⟩ is also used. |
| (,…,) | (a1, a2, …, an) | ordered n-tuple | ⟨a1, a2, …, an⟩ is also used. |
| × | A × B | cartesian product of A and B | The set of ordered pairs (a, b) such that a ∈ A and b ∈ B. A × B = { (a, b) ∣ a ∈ A ∧ b ∈ B } A × A × ⋯ × A is denoted by An, where n is the number of factors in the product. |
| Δ | ΔA | set of pairs (a, a) ∈ A × A where a ∈ A; diagonal of the set A × A | ΔA = { (a, a) ∣ a ∈ A } idA is also used. |
Read more about this topic: ISO 31-11
Famous quotes containing the word sets:
“There is the name and the thing; the name is a sound which sets a mark on and denotes the thing. The name is no part of the thing nor of the substance; it is an extraneous piece added to the thing, and outside of it.”
—Michel de Montaigne (1533–1592)
“It is odd but agitation or contest of any kind gives a rebound to my spirits and sets me up for a time.”
—George Gordon Noel Byron (1788–1824)
“We love the indomitable bellicose patriotism that sets you apart; we love the national pride that guides your muscularly courageous race; we love the potent individualism that doesn’t prevent you from opening your arms to individualists of every land, whether libertarians or anarchists.”
—Tommaso Marinetti (1876–1944)