Show that ~ is an equivalence relation

• MHB
• mathmari
In summary, the conversation discusses the equivalence relation $\sim$ defined on the set $M=\{1,2,\ldots,10\}$ using the set of subsets $\mathcal{P}=\{\{1,3,4\}, \{2,8\}, \{7\}, \{5, 6, 9, 10\}\}$. The relation is shown to be reflexive, symmetric, and transitive, making it an equivalence relation. It is also mentioned that this type of relation can be generalized for any sets and functions.
mathmari
Gold Member
MHB
Hey!

Let $M:=\{1, 2, \ldots, 10\}$ and $\mathcal{P}:=\{\{1,3,4\}, \{2,8\}, \{7\}, \{5, 6, 9, 10\}\}$.

For $x \in M$ let $[x]$ be the unique set of $\mathcal{P}$ that contains $x$.

We define the relation on $M$ as $x\sim y:\iff [x]=[y]$.

Show that $\sim$ is an equivalence relation.
For that we have to show that the relation is reflexive, symmetric and transitive.
• Reflexivity:

Let $x \in M$. Then it holds, trivially, that $[x]=[x]$. Therefore $x\sim x$. So $\sim$ is reflexive.
• Symmetry:

Let $x,y \in M$ and $x\sim y$. Then $[x]=[y]$. Equivalently it holds that $[y]=[x]$ and therefore $y \sim x$. So $\sim$ is symmetric.
• Transitivity:

Let $x,y,z\in M$ and $x\sim y$ and $y\sim z$. Then it holds that $[x]=[y]$ and $[y]=[z]$. So we have that $[x]=[y]=[z]$, so $[x]=[z]$ and therefore $x\sim z$. So $\sim$ is transitive.

Is everything correct and complete? Or do we have to justify each property with more details, i.e. using the definition of $[x]$ ? (Wondering)

Last edited by a moderator:
Well, how obvious to you consider each of your claims?

It might be useful to specify that [1]= [3]= [4]= {1, 3, 4}, that [2]= [8]= {2, 8}, that [7]= {7}, and that [5]= [6]= [9]= [10]= {5, 6, 9, 10}.

More generally, for all sets $A$ and $B$ and functions $f:A\to B$ the relation $x\sim y\iff f(x)=f(y)$ is an equivalence relation on $A$.

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between two elements in a set. It is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity.

2. How do you show that a relation is an equivalence relation?

To show that a relation is an equivalence relation, you must prove that it satisfies the three properties of reflexivity, symmetry, and transitivity. This can be done by providing specific examples or using logical proofs.

3. What is the importance of equivalence relations in science?

Equivalence relations are important in science because they allow us to classify and compare objects or phenomena based on their properties or characteristics. This helps us to better understand and analyze complex systems and relationships.

4. Can you give an example of an equivalence relation in science?

One example of an equivalence relation in science is the concept of isomorphism. In chemistry, isomorphism is used to describe compounds with the same chemical formula but different structures. This relation is reflexive, symmetric, and transitive.

5. How is an equivalence relation different from other types of relations?

An equivalence relation differs from other types of relations, such as partial orders or total orders, because it does not impose any kind of ordering or hierarchy on the elements in a set. Instead, it focuses on the equality or similarity between elements.

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