Isomorphism as an Equivalence Relation on Sets: A Proof

In summary, the conversation discusses proving that isomorphism is an equivalence relation on a set of sets, S. The individual proposes using the Reflexive, Symmetrical, and Transitive properties to approach the proof. The other individual confirms that these steps are necessary and clarifies that X! refers to the set of all selfmap functions on set X.
  • #1
yaganon
17
0
So it says here "Let S be a set of sets. Show that isomorphism is an equivalence relation on S."

So in order to approach this proof, can I just use the Reflexive, Symmetrical, and Transitive properties that is basically the definition of equivalence relations?

eg. suppose x, y, z are sets contained in S...

Reflexive: x~x <same as> xRx
Symmetry: x~y => y~x <same as> xRy => yRx
Transitive: x~y, y~z => x~z

This seems too elementary, but I just want to make sure.
 
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  • #2
Yes - showing something is an equivalence relation always involves the same steps. Sets are isomorphic if they have the same cardinality, so you need to show that the property of "having the same number of elements" satisfies the three conditions.
 
  • #3
I'm also confused about the X! notation. I think X^x means the set of all selfmap functions on X. What is X! suppose to be?
 

1. What is a set of sets?

A set of sets is a collection of sets, where each set within the collection is considered an element of the larger set. For example, a set of sets may include the sets {1, 2, 3}, {a, b, c}, and {dog, cat, bird}.

2. What is an isomorphism?

An isomorphism is a mathematical concept that describes a bijective function between two mathematical objects that preserves their structure. In the context of sets, an isomorphism between two sets means that they have the same number of elements and their elements are paired in a one-to-one correspondence.

3. How is an isomorphism different from an equivalence relation?

An isomorphism is a more specific type of equivalence relation. While both are binary relations that establish a connection between two mathematical objects, an isomorphism also requires that the structure of the objects is preserved. In other words, an isomorphism is an equivalence relation that also has a specific function mapping the elements of one object to the other.

4. Why is the concept of isomorphism important in mathematics?

The concept of isomorphism is important in mathematics because it allows us to compare and contrast different mathematical structures and objects. It also helps us to identify patterns and similarities between seemingly different objects, which can lead to deeper insights and understanding of mathematical concepts.

5. Can an isomorphism exist between sets of different sizes?

No, an isomorphism can only exist between sets of the same size. This is because an isomorphism requires a one-to-one correspondence between the elements of two sets, which is not possible if the sets have different numbers of elements.

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