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

• MHB
• leigh ramona
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.
leigh ramona
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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First, do you know what "reflexive", "symmetric", and "transitive" mean? Write down the definitions and show that this relation satisfies those definitions.

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.

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.

## 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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