离散数学这题如何解
1、设集合 A={1,2,3,4,5},A 上等价关系 R 定义为:
R={⟨1,2⟩,⟨2,1⟩,⟨3,4⟩,⟨4,3⟩}∪IA
A) 求 R 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} and the equivalence relation R defined as R={⟨1,2⟩,⟨2,1⟩,⟨3,4⟩,⟨4,3⟩}∪IA, where IA is the identity relation on A
step 2
The identity relation IA on A is IA={⟨1,1⟩,⟨2,2⟩,⟨3,3⟩,⟨4,4⟩,⟨5,5⟩}
step 3
Combining R and IA, we get R={⟨1,2⟩,⟨2,1⟩,⟨3,4⟩,⟨4,3⟩,⟨1,1⟩,⟨2,2⟩,⟨3,3⟩,⟨4,4⟩,⟨5,5⟩}
step 4
To find the equivalence classes, we group elements that are related by R
step 5
The equivalence class of 1 is [1]={1,2} because ⟨1,2⟩ and ⟨2,1⟩ are in R
step 6
The equivalence class of 2 is [2]={1,2} because ⟨1,2⟩ and ⟨2,1⟩ are in R
step 7
The equivalence class of 3 is [3]={3,4} because ⟨3,4⟩ and ⟨4,3⟩ are in R
step 8
The equivalence class of 4 is [4]={3,4} because ⟨3,4⟩ and ⟨4,3⟩ are in R
step 9
The equivalence class of 5 is [5]={5} because ⟨5,5⟩ is in R
Answer
The equivalence classes of R are {1,2}, {3,4}, and {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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