```
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:
$R=\{\langle 1,2\rangle,\langle 2,1\rangle,\langle 3,4\rangle,\langle 4,3\rangle\} \cup I_{A}$
A) 求 $R$ 的所有等价类。
```

Sign in to unlock the answer

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 $R = \{\langle 1, 2 \rangle, \langle 2, 1 \rangle, \langle 3, 4 \rangle, \langle 4, 3 \rangle\} \cup I_A$, where $I_A$ is the identity relation on $A$

step 2

The identity relation $I_A$ on $A$ is $I_A = \{\langle 1, 1 \rangle, \langle 2, 2 \rangle, \langle 3, 3 \rangle, \langle 4, 4 \rangle, \langle 5, 5 \rangle\}$

step 3

Combining $R$ and $I_A$, we get $R = \{\langle 1, 2 \rangle, \langle 2, 1 \rangle, \langle 3, 4 \rangle, \langle 4, 3 \rangle, \langle 1, 1 \rangle, \langle 2, 2 \rangle, \langle 3, 3 \rangle, \langle 4, 4 \rangle, \langle 5, 5 \rangle\}$

step 4

To find the equivalence classes, we group elements that are related by $R$

step 5

The equivalence class of 1 is $[1] = \{1, 2\}$ because $\langle 1, 2 \rangle$ and $\langle 2, 1 \rangle$ are in $R$

step 6

The equivalence class of 2 is $[2] = \{1, 2\}$ because $\langle 1, 2 \rangle$ and $\langle 2, 1 \rangle$ are in $R$

step 7

The equivalence class of 3 is $[3] = \{3, 4\}$ because $\langle 3, 4 \rangle$ and $\langle 4, 3 \rangle$ are in $R$

step 8

The equivalence class of 4 is $[4] = \{3, 4\}$ because $\langle 3, 4 \rangle$ and $\langle 4, 3 \rangle$ are in $R$

step 9

The equivalence class of 5 is $[5] = \{5\}$ because $\langle 5, 5 \rangle$ 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.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Unlock Smarter Learning with AskSia Super!

Join Super, our all-in-one AI solution that can greatly improve your learning efficiency.

30% higher accuracy than GPT-4o

Entire learning journey support

The most student-friendly features

Study Other Question