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An extensional definition gives meaning to a term by specifying its extension, that is, every object that falls under the definition of the term in question.. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class.
A left identity element that is also a right identity element if called an identity element. The empty set ∅ {\displaystyle \varnothing } is an identity element of binary union ∪ {\displaystyle \cup } and symmetric difference , {\displaystyle \triangle ,} and it is also a right identity element of set subtraction ∖ : {\displaystyle ...
If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
Letting a particular isomorphism identify the two structures turns this heap into a group. In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations.
Use specific wording rather than vague whenever possible: For example, instead of saying, "in the old days", specify the time period you are referring to (say, the 19th century, or the 1960s). Similarly, do not reference "now": the word "Recently" or "Soon" or their synonyms will become meaningless quickly. Use "As of" or a specific date.
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So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".