Proving Equivalence Relations: Intersection of Two Sets

In summary, the conversation discusses the topic of equivalence relations and whether the intersection of two equivalence relations is itself an equivalence relation. The participants also mention the need for showing their work and discussing definitions in order to receive help.
  • #1
nishap
1
0
Hi All

I have a problem with Set theory. I am given to prove the following;

Is the intersection of two equivalence relations itself an equivalance relation? If so , how would you characterize the equivalnce sets of the intersection?

Regards,
Nisha.
 
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  • #2
Hi Nisha,

We have a policy here which is posted at the top of this Forum. In order to receive help, you have to show us how you started and where you got stuck.
 
  • #3
For any sets A and B, a (binary) relation from A to B is a subset of AxB.

Of what set is an equivalence relation a subset?

What conditions must this subset satisfy?

Since equivalence relations are (sub)sets, the intersection of 2 equivalence relations is a set. Does this set satisfy the conditions required of an equivalence relation?

Regards,
George
 
  • #4
Just in case you have no clue where to start - going back to the definitions is very helpful.
 

Related to Proving Equivalence Relations: Intersection of Two Sets

1. What is the definition of an equivalence relation?

An equivalence relation is a mathematical concept that describes a relationship between two elements in a set. It states that two elements are considered equivalent if they share certain properties or characteristics, such as having the same value or satisfying the same conditions.

2. How do you prove that two sets are equivalent?

To prove that two sets are equivalent, you must show that they have the same cardinality (number of elements) and that each element in one set corresponds to one and only one element in the other set. This can be done by establishing a one-to-one mapping between the elements of the two sets.

3. What is the intersection of two sets?

The intersection of two sets is the set of elements that are common to both sets. In other words, it is the overlap between the two sets - the elements that they have in common.

4. How do you prove that the intersection of two sets is an equivalence relation?

To prove that the intersection of two sets is an equivalence relation, you must show that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity. This means that the intersection of a set with itself is the same set, the intersection is commutative, and the intersection of two equivalent sets is also equivalent.

5. What are some examples of using the intersection of two sets to prove equivalence?

One example is proving that two triangles are congruent by showing that their corresponding angles and sides are equal. Another example is proving that two matrices are similar by showing that their eigenvectors and eigenvalues are the same.

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