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aaaa202
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Is 0 I am told. Is this an axiom, or can it be proven?
mathman said:By definition, the cardinality of any finite set is the number of elements.
economicsnerd said:In our case, there is a bijection between [itex]\emptyset[/itex] and [itex]0=\emptyset[/itex], so we're good.
economicsnerd said:There's a collection of sets which are called cardinal sets. I won't define a cardinal in general, but the finite ones are defined like:
- [itex]0:=\emptyset[/itex].
aaaa202 said:Let me try to be more precise about what worries me about the cardinality of the empty set. A set X is countable if there exists an injection from X to N. So is the empty set countable?
economicsnerd said:There's a collection of sets which are called cardinal sets. I won't define a cardinal in general, but the finite ones are defined like:
- [itex]0:=\emptyset[/itex]
- [itex]1:= 0 \cup \{0\} = \{\emptyset\}[/itex]
-[itex]2:= 1 \cup \{1\} = \{\emptyset, \{\emptyset\} \}[/itex]
...
-[itex]k+1:=k\cup \{k\}[/itex]
We name these things like numbers, but they're just sets like any other.
By definition, a set [itex]A[/itex] has cardinality [itex]\kappa[/itex] if [itex]\kappa[/itex] is a cardinal set and there exists a bijection between [itex]A[/itex] and [itex]\kappa[/itex]. In our case, there is a bijection between [itex]\emptyset[/itex] and [itex]0=\emptyset[/itex], so we're good.
MrAnchovy said:Neither; it is part of the definition of cardinality (the cardinality of the empty set is defined to be 0). If it was defined to be any other number, or left undefined, then (among other problems) the equality ## \mid A \cup B \mid = \mid A \mid + \mid B \mid - \mid A \cap B \mid ## would not hold if ## A ## or ## B ## is the empty set.
willem2 said:This equation ALWAYS holds if A or B is the empty set, no matter how caridinality is defined for ANY set
Zafa Pi said:I have two questions:
1. Why doesn't $$1\cup \{1\} = \{\{\emptyset\}, \{\{\emptyset\}\} \}?$$
2. Given MrAnchovy's response, are you happy with the confusion you've sown?
The cardinality of an empty set is 0. This means that the empty set contains no elements.
The cardinality of an empty set is determined by counting the number of elements in the set. Since the empty set has no elements, its cardinality is 0.
Yes, the cardinality of an empty set is always 0. This is because the empty set contains no elements, so there is nothing to count.
No, an empty set can only have a cardinality of 0. This is a fundamental mathematical property of the empty set.
The cardinality of an empty set is important because it is used as a basis for defining other sets, such as the natural numbers. It also plays a role in set operations and can help clarify the differences between finite and infinite sets.