Below you will find example sentences with "empty set". The examples show how this phrase is used in natural context and which words often surround it.
Empty Set in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: set
- Corpus frequency in the collocation scan: 22
- Phrase length: 2 words
- Average sentence length: 28.5 words
Sentence profile
- Phrase position: 9 start, 9 middle, 2 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "empty set" has 2 words and usually appears near the start in these examples. The average sentence has 28.5 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as is the empty set, in the empty set and the, elements, open and element stand out.
- In the phrase index, this combination connects with instruction set, set aside, set theory and set theory, linking the page to nearby combinations.
Example types with empty set
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
The closure of the empty set is empty. (8 words)
The empty set serves as the initial object in Set with empty functions as morphisms. (15 words)
The empty set equipped with the empty binary operation satisfies this definition of a quasigroup. (15 words)
In other words, the power set of the empty set is the set containing the empty set and the power set of any other set is all the subsets of the set containing some specific element and all the subsets of the set not containing that specific element. (48 words)
All its boundary points (of which there are none) are in the empty set, and the set is therefore closed; while for every one of its points (of which there are again none), there is an open neighbourhood in the empty set, and the set is therefore open. (48 words)
The ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty. (45 words)
Example sentences (20)
In other words, the power set of the empty set is the set containing the empty set and the power set of any other set is all the subsets of the set containing some specific element and all the subsets of the set not containing that specific element.
The ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.
Hence there is but one empty set, and we speak of "the empty set" rather than "an empty set".
All its boundary points (of which there are none) are in the empty set, and the set is therefore closed; while for every one of its points (of which there are again none), there is an open neighbourhood in the empty set, and the set is therefore open.
It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself. citation.
Note that the empty set is a subset of every set (the statement that all elements of the empty set are also members of any set A is vacuously true ).
Operations on the empty set When speaking of the sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set is zero.
Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set ; in other theories, its existence can be deduced.
When this is done, the empty set is the Von Neumann cardinal assignment for a set with no elements, which is the empty set.
Examples a is an interior point of M, because there is an ε-neighbourhood of a which is a subset of M. *In any space, the interior of the empty set is the empty set.
The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements.
The empty set serves as the initial object in Set with empty functions as morphisms.
Such sets include the * Smith set : The smallest non-empty set of candidates in a particular election such that every candidate in the set can beat all candidates outside the set.
Similary, the product of the elements of the empty set should be considered to be one (see empty product ), since one is the identity element for multiplication.
The closure of the empty set is empty.
The empty set equipped with the empty binary operation satisfies this definition of a quasigroup.
Begin with a formula θ(C) that does not mention B, and a set A. If no element E of A satisfies θ then the set B desired by the relevant instance of the axiom schema of separation is the empty set.
Moreover, the empty set is a compact set by the fact that every finite set is compact.
The empty set and the set of all reals are open intervals, while the set of non-negative reals, for example, is a right-open but not left-open interval.
The empty set is not the same thing as nothing ; rather, it is a set with nothing inside it and a set is always something.