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zwqqzeng's Question
Math
Posted 4 months ago
离散数学这题如何解
1、设集合 A={1,2,3,4,5},AA=\{1,2,3,4,5\}, A 上等价关系 RR 定义为:
R={1,2,2,1,3,4,4,3}IA
R=\{\langle 1,2\rangle,\langle 2,1\rangle,\langle 3,4\rangle,\langle 4,3\rangle\} \cup I_{A}

A) 求 RR 的所有等价类。
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Answer from Sia
Posted 4 months ago
Solution by Steps
step 1
Given the set A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} and the equivalence relation RR defined as R={1,2,2,1,3,4,4,3}IAR = \{\langle 1, 2 \rangle, \langle 2, 1 \rangle, \langle 3, 4 \rangle, \langle 4, 3 \rangle\} \cup I_A, where IAI_A is the identity relation on AA
step 2
The identity relation IAI_A on AA is IA={1,1,2,2,3,3,4,4,5,5}I_A = \{\langle 1, 1 \rangle, \langle 2, 2 \rangle, \langle 3, 3 \rangle, \langle 4, 4 \rangle, \langle 5, 5 \rangle\}
step 3
Combining RR and IAI_A, we get R={1,2,2,1,3,4,4,3,1,1,2,2,3,3,4,4,5,5}R = \{\langle 1, 2 \rangle, \langle 2, 1 \rangle, \langle 3, 4 \rangle, \langle 4, 3 \rangle, \langle 1, 1 \rangle, \langle 2, 2 \rangle, \langle 3, 3 \rangle, \langle 4, 4 \rangle, \langle 5, 5 \rangle\}
step 4
To find the equivalence classes, we group elements that are related by RR
step 5
The equivalence class of 1 is [1]={1,2}[1] = \{1, 2\} because 1,2\langle 1, 2 \rangle and 2,1\langle 2, 1 \rangle are in RR
step 6
The equivalence class of 2 is [2]={1,2}[2] = \{1, 2\} because 1,2\langle 1, 2 \rangle and 2,1\langle 2, 1 \rangle are in RR
step 7
The equivalence class of 3 is [3]={3,4}[3] = \{3, 4\} because 3,4\langle 3, 4 \rangle and 4,3\langle 4, 3 \rangle are in RR
step 8
The equivalence class of 4 is [4]={3,4}[4] = \{3, 4\} because 3,4\langle 3, 4 \rangle and 4,3\langle 4, 3 \rangle are in RR
step 9
The equivalence class of 5 is [5]={5}[5] = \{5\} because 5,5\langle 5, 5 \rangle is in RR
Answer
The equivalence classes of RR are {1,2}\{1, 2\}, {3,4}\{3, 4\}, and {5}\{5\}.
Key Concept
Equivalence Classes
Explanation
Equivalence classes group elements that are related by an equivalence relation. In this case, elements 1 and 2 are related, 3 and 4 are related, and 5 is only related to itself.

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