Proving Reflexive, Symmetric and Transitive Properties of Relation R on P(U)

In summary, the conversation discusses a relation $R$ defined on the power set of a universal set $U$ where $A R B$ holds if $A\cap C = B\cap C$. The task is to determine if $R$ is reflexive, symmetric, and/or transitive and to prove the answer. The definitions of reflexive, symmetric, and transitive are discussed, along with the need for precise language when discussing mathematical concepts. It is determined that $R$ is symmetric and transitive, but it is unclear if it is reflexive without more information.
  • #1
leigh ramona
2
0
Let U be a universal set, and let C be any subset of U. Let R be the relation on P(U) defined by A R B if $A \cap C = B \cap C$. Determine whether the relation is reflexive, symmetric, and/or transitive. Prove you answer.
 
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  • #2
First, do you know what "reflexive", "symmetric", and "transitive" mean? Write down the definitions and show that this relation satisfies those definitions.
 
  • #3
So reflexive is equal to each other. Like x R x.
Symmetric is x R y = y R x
Transitive is if x R y and y R z, then x R z.

The relation is symmetric because if A \cap C = B \cap C, then C \cap A = C \cap B. Is this correct?

The relation is also transitive, because if A \cap C and B \cap C, then A \cap B.

I'm not sure about the reflexive.
 
  • #4
leigh ramona said:
So reflexive is equal to each other. Like x R x.
Your sentence does not make sense because it lacks the subject: what is equal to each other?. And when you say $xRx$, I'll ask you: what is $x$? Do you mean $xRx$ holds for some unspecified $x$, for all $x$, for some specific $x$? What set does $x$ range over?

leigh ramona said:
Symmetric is x R y = y R x
This sentence is also problematic. For each $x$ and $y$, $xRy$ is either true or false. What do you mean by $xRx=yRx$? For which $x$ and $y$?

leigh ramona said:
Transitive is if x R y and y R z, then x R z.
This would be correct if you added "for all $x$, $y$ and $z$".

leigh ramona said:
The relation is symmetric because if A \cap C = B \cap C, then C \cap A = C \cap B. Is this correct?
What you wrote is true, but what does this have to do with $R$? Please enclose formulas in dollar signs: \$A\cap C\$.

leigh ramona said:
The relation is also transitive, because if A \cap C and B \cap C, then A \cap B.
This does not makes sense because $A\cap C$ cannot be true or false: it's a set. Therefore, you can't write "If $A\cap C$...".

If you increase the level of your precision, it will help you not only to communicate more clearly, but to understand the problem and definitions better as well.
 

Related to Proving Reflexive, Symmetric and Transitive Properties of Relation R on P(U)

1. What is the definition of a reflexive relation?

A relation R on a set U is reflexive if every element in U is related to itself, i.e. (a,a) ∈ R for all a ∈ U.

2. How can we prove that a relation R on P(U) is reflexive?

To prove that R on P(U) is reflexive, we need to show that for every subset A ∈ P(U), (A,A) ∈ R. This can be done by using the definition of reflexive relation and substituting A for a in the definition.

3. What is the symmetric property of a relation?

A relation R on a set U is symmetric if for any elements a and b in U, (a,b) ∈ R implies (b,a) ∈ R.

4. How can we prove the symmetric property of a relation R on P(U)?

To prove that R on P(U) is symmetric, we can show that if (A,B) ∈ R, then (B,A) ∈ R for any subsets A and B in P(U). This can be done by using the definition of symmetric relation and substituting A and B for a and b in the definition.

5. What is the transitive property of a relation?

A relation R on a set U is transitive if for any elements a, b, and c in U, if (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R.

6. How can we prove the transitive property of a relation R on P(U)?

To prove that R on P(U) is transitive, we can show that if (A,B) ∈ R and (B,C) ∈ R, then (A,C) ∈ R for any subsets A, B, and C in P(U). This can be done by using the definition of transitive relation and substituting A, B, and C for a, b, and c in the definition.

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