Symmetric/Antisymmetric Relations, Set Theory Problem

In summary, the conversation is discussing the validity of the claim that if R is a symmetric relation on A, and Dom(R) = A, then R = the identity relation. The person does not believe the claim and provides a counterexample. They also suggest that the claim might actually mean that R should be both symmetric and antisymmetric in order for it to be equal to the identity relation. However, they point out that any equivalence relation is symmetric and has domain A, so the claim cannot be true.
  • #1
alec_tronn
29
0

Homework Statement


Prove that if R is a symmetric relation on A, and Dom(R) = A, then R = the identity relation.


2. The attempt at a solution
My problem is... I don't believe the claim. At all. If A = {1, 2, 3} and R = {(1, 2), (2, 1), (3, 1), (1, 3)}, that satisfies the antecedent, and isn't the identity relation. Am I missing something? I can't exactly prove something I don't believe. Thanks for any help or explanations you can provide.

p.s. This book has been known to have typos eeeeeverywhere. Suggestions as to what they really meant (antisymmetric? that wouldn't even work I don't think) are appreciated as well.
 
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  • #2
What exactly do you mean by Dom(R)=A?
 
  • #3
Domain of R is the entire set A.
 
  • #4
I don't know what they could mean. Take any relation that satisfies the premise (such as yours, or the identity relation) and add pairs of relations, keeping it symmetric, and it still satisfies the premise. It seems like the Dom(R) part is crucial (since the symmetric part is pretty straightforward), so are you sure you've interpreted this right? You probably have, I'm just not familiar with that notation, and I don't know what else to suggest.
 
  • #5
I've decided that it'd be easier to prove that if R is symmetric and antisymmetric R = identity relation (it's the only thing I can think of). That should hold true shouldn't it?
 
  • #6
Yea, and the domain still must be all of A.
 
  • #7
Gah, whoever uses subsets to define relations should be taken outside and have their maths qualifications thoroughly slapped.

Any equivalence relation is symmetric, and has domain A, surely. So the claim can't be true.
If we have a relation that is reflexive (a~a) but satisfies a does not ~ any other b, then yes, only the identity satisfies that.
 

1. What is a symmetric relation?

A symmetric relation is a type of binary relation in which the order of the elements does not matter. In other words, if A is related to B, then B is also related to A. This property is denoted as "A ~ B" or "A R B", where "~" or "R" represents the relation.

2. How is a symmetric relation different from an antisymmetric relation?

A symmetric relation and an antisymmetric relation are two different types of binary relations. While a symmetric relation follows the property that if A is related to B, then B is also related to A, an antisymmetric relation follows the opposite property that if A is related to B, then B is not related to A. In other words, an antisymmetric relation cannot have both A ~ B and B ~ A as true statements.

3. Can a relation be both symmetric and antisymmetric?

No, a relation cannot be both symmetric and antisymmetric because these properties are contradictory. A relation can either have the symmetric property or the antisymmetric property, but not both at the same time.

4. How is set theory used in understanding symmetric and antisymmetric relations?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. In the context of symmetric and antisymmetric relations, sets are used to represent the elements involved in the relation. By using set theory concepts, we can determine if a relation is symmetric or antisymmetric by analyzing the elements and their relationships.

5. Can you give an example of a symmetric and antisymmetric relation?

One example of a symmetric relation is the "equal to" relation between numbers. For instance, 3 is equal to 3, and 4 is equal to 4, so the relation is symmetric. An example of an antisymmetric relation is the "less than or equal to" relation between numbers. For instance, 3 is less than or equal to 4, but 4 is not less than or equal to 3, so the relation is antisymmetric.

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