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:
““Yes” shuts us in. “No” sets us free.”
—Mason Cooley (b. 1927)
“Analysis as an instrument of enlightenment and civilization is good, in so far as it shatters absurd convictions, acts as a solvent upon natural prejudices, and undermines authority; good, in other words, in that it sets free, refines, humanizes, makes slaves ripe for freedom. But it is bad, very bad, in so far as it stands in the way of action, cannot shape the vital forces, maims life at its roots. Analysis can be a very unappetizing affair, as much so as death.”
—Thomas Mann (1875–1955)
“I would rather have as my patron a host of anonymous citizens digging into their own pockets for the price of a book or a magazine than a small body of enlightened and responsible men administering public funds. I would rather chance my personal vision of truth striking home here and there in the chaos of publication that exists than attempt to filter it through a few sets of official, honorably public-spirited scruples.”
—John Updike (b. 1932)