In the state already know the product set, to explore the relationship between the collection elements.
I. equivalence relation
[Definition \ (3.1 \) ]
a. provided \ (A, B \) as a set, product set \ (A \ times B \) of a subset \ (R & lt \) is called \ (A \) to \ (B \) of a relationship, in particular, he said the \ (a \ times a \) is a subset of (a \) \ a relationship on. If \ ((A, b) \ in R \) , called \ (a, b \) and \ (R \) associated, denoted by \ (aRb \) .
[Definition \ (3.2 \) ]
. b disposed \ (R & lt \) is \ (A \) on a relationship, if \ (R & lt \) satisfies the following condition:
ba own anti-resistance: young \ (A \ In A \) , Yes \ ((A, A) \ In R \) .
bb对称of: young \ ((A, B) \ In R \) , Yes \ ((B, A) \ In R \) .
b.c.传递性：若 \((a,b),(b,c)\in R\)，有 \((a,c)\in R\).
We call \ (R \) is \ (A \) an equivalence relation on, commonly used \ (\ sim \) , said the upcoming \ (aRb \) denoted \ (A \ the SIM b \) .
The Tip: \ (R & lt \) is the in \ (A \ times B \) plus a number of constraints, such as provided set \ (A, B \) , \ (A \ B = Times \ {(X, Y) | X \ in a, Y \ in B \} \) , \ (R & lt \) formula can be written as \ (R = \ {(x , y) \ in a \ times B, P (x, y) \} \) , where \ (P (x, y) \) of \ (R & lt \) constraints.
Example 1 is provided \ (D \) is \ (Descartes are \) plane defined: \ (D \) two points \ (a \ sim b \) if and only if \ (a, b \) distance from the origin equal to . The \ (P (a, b) \) i.e. \ (a, b \) to the origin equal distance, this is not difficult to verify equivalence relation.
Example 2 set \ (H \) is the set of all human beings, is defined: \ (H \) of the two elements \ (a \ sim b \) if and only if \ (a, b \) of the same sex. The \ (P (a, b) \) i.e. \ (a, b \) of the same sex is also equivalent relation.
Tip: As if and only if no classification (necessary and sufficient) discuss only need to verify in this definition \ (R & lt \) is equivalent conditions can.
II. Partition
a. are located \ (\ SIM \) is set \ (A \) an equivalence relation on, \ (A \) of \ (A \) within an element, and \ (A \) in \ (\ SIM ) \ equivalent of all the elements \ (a \) a subset, referred to as \ (a \) in an equivalence class with \ ([a] \) FIG.
In b. Example. 1 \ (A \) of the equivalence class is (A \) \ set composed of all points on the same center as the origin of the circle, in Example 2 \ (H \) contain only two other price categories, respectively, for men and women. Note the difference of expression.
c. I note, within a set of mutually different elements where two equivalence class if coincident, certainly do not intersect. I.e., each independently of each equivalence class, equivalence class twenty-two intersection is empty , the whole and is set \ (A \) .
. E Thus we see:
If a set ea defines an equivalence relation on, then the set may be divided into sub-sets of disjoint and.
eb is represented if a sub-set of disjoint sets of and, called a partition for a subset of these aromatic set.
In fact, we can derive the following proposition:
[Proposition \ (3.1 \) ]
1. set \ (R & lt \) is set (A \) \ an equivalence relation on, the \ (R & lt \) determines the \ (A \) to a reticle \ (P \) , and by the \ (P \) derived equivalence relation is the \ (R \) .
2. Given \ (A \) to a reticle \ (P \) , can also derive a \ (A \) equivalence relation on \ (R & lt \) , and by the \ (R & lt \) partition determined that is, \ (P \) .
Proof omitted (after all, this is just a note).
III. Quotient set
a. provided \ (\ SIM \) is set \ (A \) an equivalence relation on, \ (A \) group all equivalence classes of the set is called \ (A \) on \ (\ SIM \) the quotient set, referred to as the \ (a / \ sim \) or \ (\ overline a \) .
b. If \ (A \ in A \) , then \ ([A] \) as \ (\ overline A \) of the elements are usually referred to as \ (\ overline A \) .