Subset
In mathematics, a set B is a subset of another set A if all elements of B also belong to A . We then say that B "is contained" within A.
Definition
The difference between the sets is formed by elements that belong to one and not to the others.
Other ways of saying it are “B is included in A”, “A includes B”, etc.
Examples.
- The “joint of all women” is a subset of the “joint of all people”.
- {1,3} {1, 2, 3, 4!
- {2, 4, 6,...} {1, 2, 3,} = N
({Even numbers} ⊆ {Natural numbers})
Subset pairs
It is true that every element of a set A is an element of A (it is a tautological statement). Therefore, we have the following theorem:
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Thus, given two sets A ⊆ B, it is possible that they are the same, A = B.
On the other hand, it is also possible that A contains some but not all elements of B:
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It is true that all the subset examples shown above are in fact proper subsets.
The notation A ⊂ B and B ⊃ A, but according to the author this can denote subset, A ⊆ B and B ⊇ A; or proper subset, A ⊊ B and B ⊋ A.
Power set
The totality of the subsets of a given set A constitutes the so-called power set or parts set of A:
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When the set A has a finite number of elements, for example |A| = n, the power set is also finite and has 2n items.
Example. Given the set A = {a, b }, its power set is:
- P(A)={∅ ∅ ,{a!,{b!,{a,b!!{displaystyle {mathcal {P}(A)={emptyset,{a},{b}},,{a,b}}}}
Properties
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This is because «every ∅ element is A” means the same as “∅ does not have any elements that are in A”, and this is true regardless of A since ∅ has no elements.
If each element of a set A is an element of another set B , and each element of B is in turn an element of another set C, then each member of A also belongs to C, that is:
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Furthermore, if two sets are subsets of each other, then all members of one are of the other and vice versa. Then, both sets have the same elements, and the sets are defined only by their elements, then:
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Advanced properties
The include relation has the same properties as the loose order relation: it is reflexive (A ⊆ A); transitive (A ⊆ B and B ⊆ C imply A ⊆ C); and antisymmetric (A ⊂ B and B ⊂ A imply A = B).
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