# Reflexive, Symmetric, Transitive - Prove related problem

• eseefreak
In summary: For example, if you are to prove that P is reflexive, you need to show that for any (x,y) in A, (x,y) P (x,y). Substitute (x,y) for (z,w) in the original definition, and you should get (x,y) P (x,y) as the result.To show that P is symmetric, you need to show that for any (x,y) and (z,w) in A, if (x,y) P (z,w) then (z,w) P (x,y). Substitute (x,y) and (z,w) for (x,y) and (z,w) in the original definition, and you should get (z,w) P (
eseefreak

## Homework Statement

Let A=RxR=the set of all ordered pairs (x,y), where x and y are real numbers. Define relation P on A as follows: For all (x,y) and (z,w) in A, (x,y)P(z,w) iff x-y=z-w

## Homework Equations

R is reflexive if, and only if, for all x ∈ A,x R x.
R is symmetric if, and only if, for all x,y∈A,if xRy then yRx.
R is transitive if, and only if, for all x,y,z∈A, if xRy and yRz then xRz.

## The Attempt at a Solution

I am supposed to prove that P is reflexive, symmetric and transitive.
To show if P is reflexive do I just state that since y-x=w-z then L is reflexive
To show it is symmetric do I just state y-x=w-z and x-y=z-w, since they are equivalent then they are symmetric
To show if Transitive if (the book gives this answer for a similar problem) since x-y and z-w are integers then x-z=(x-y) + (x-z) is the sum of two integers. Therefore x-z is an integer. Assuming this is correct, are my two answers for reflexive and symmetric incorrect?

Furthermore, how would I list five elements in [(2,6)] or [(5,5)]?

Thanks for the help! :)

eseefreak said:

## Homework Statement

Let A=RxR=the set of all ordered pairs (x,y), where x and y are real numbers. Define relation P on A as follows: For all (x,y) and (z,w) in A, (x,y)P(z,w) iff x-y=z-w

## Homework Equations

R is reflexive if, and only if, for all x ∈ A,x R x.
R is symmetric if, and only if, for all x,y∈A,if xRy then yRx.
R is transitive if, and only if, for all x,y,z∈A, if xRy and yRz then xRz.

## The Attempt at a Solution

I am supposed to prove that P is reflexive, symmetric and transitive.
To show if P is reflexive do I just state that since y-x=w-z then L is reflexive
What do you get for (x, y)P(x, y)?
eseefreak said:
To show it is symmetric do I just state y-x=w-z and x-y=z-w, since they are equivalent then they are symmetric
You need to show that (x, y) P (z, w) is the same as (z, w) P (x, y)
eseefreak said:
To show if Transitive if (the book gives this answer for a similar problem) since x-y and z-w are integers then x-z=(x-y) + (x-z) is the sum of two integers.
You can't assume the numbers are integers. The relations are defined on sets of real ordered pairs.
eseefreak said:
Therefore x-z is an integer. Assuming this is correct, are my two answers for reflexive and symmetric incorrect?

Furthermore, how would I list five elements in [(2,6)] or [(5,5)]?
Use the definition of your relation.
eseefreak said:
Thanks for the help! :)

Last edited:
Mark44 said:
What do you get for (x, y)P(x, y)?
You need to show that (x, y) P (z, w) is the same as (z, w) P (x, y)

Use the definition of your relation.

How can I show that they are the same? It seems easy to show they are the same if there are variables in place but how would I show they are the same for all real numbers?

btw thanks Mark44 for the help

What is (z,w)P(x,y)? my guess would be having to prove that x-y=y-x

Just say "Let w,x,y,z be arbitrary elements of A such that (x,y)P(z,w)". Then prove that (z,w)P(x,y), without making any assumptions about w,x,y,z.

eseefreak said:
What is (z,w)P(x,y)? my guess would be having to prove that x-y=y-x
The original definition is this: (x, y) P (z, w) iff x - y = z - w.
(x, y) P (z, w) and x - y = z - w are statements with identical truth values. If either one is true, the other is true. If either one is false, the other is false.For the relation P, the ordered pairs (x, y) and (z, w) satisfy the relation if and only if x - y = z - w.

For the various parts of this problem, substitute the given variables in the equation on the right, making sure to put them in the right places.

## 1. What do reflexive, symmetric, and transitive mean in relation to mathematical proofs?

Reflexive, symmetric, and transitive refer to properties of relations between elements in a set. In a reflexive relation, every element is related to itself. In a symmetric relation, if element A is related to element B, then element B is also related to element A. In a transitive relation, if element A is related to element B and element B is related to element C, then element A is also related to element C.

## 2. How do you prove that a relation is reflexive?

To prove that a relation is reflexive, you must show that every element in the set is related to itself. This can be done by showing that for every element a in the set, (a, a) is an element of the relation.

## 3. What is an example of a symmetric relation?

An example of a symmetric relation is the "equal to" relation. If a = b, then b = a. This property is also seen in the "is parallel to" relation. If line AB is parallel to line CD, then line CD is also parallel to line AB.

## 4. How can you prove that a relation is transitive?

To prove that a relation is transitive, you must show that if element A is related to element B and element B is related to element C, then element A is also related to element C. This can be done by showing that for every element a, b, and c in the set, if (a, b) and (b, c) are both elements of the relation, then (a, c) is also an element of the relation.

## 5. What is the importance of understanding reflexive, symmetric, and transitive properties in mathematics?

Understanding these properties is crucial in mathematical proofs and in understanding the properties of different relations. These properties help to establish relationships between elements in a set and can be used to prove or disprove certain statements or theorems.

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