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:
“Whether changes in the sibling relationship during adolescence create long-term rifts that spill over into adulthood depends upon the ability of brothers and sisters to constantly redefine their connection. Siblings either learn to accept one another as independent individuals with their own sets of values and behaviors or cling to the shadow of the brother and sister they once knew.”
—Jane Mersky Leder (20th century)
“Drink, sir, is a great provoker of three things ... nose-painting, sleep, and urine. Lechery, sir, it provokes and unprovokes: it provokes the desire but it takes away the performance. Therefore much drink may be said to be an equivocator with lechery: it makes him and it mars him; it sets him on and it takes him off.”
—William Shakespeare (1564–1616)
“There be some sports are painful, and their labor
Delight in them sets off. Some kinds of baseness
Are nobly undergone, and most poor matters
Point to rich ends.”
—William Shakespeare (1564–1616)