# Proving Relation Properties: A Guide

• BMath
In summary, the conversation discusses a question about proving that a given relation is reflexive, symmetric, and transitive. The participants provide a definition of the relation and discuss how to prove each property. The conversation also includes examples and counterexamples to help understand the concepts. In the end, it is determined that the relation is not an equivalence relation and the importance of proving properties in general is emphasized.

#### BMath

Hey,

Ive a question in my textbook and I don't really know what to do!

The question is:

I know I need to prove that the relation is
1)Symmetric
2)Reflexive
3)Transitive
But how do I prove this

Do you know what reflexivity, symmetry, and transitivity mean?

Yes and this is my proof:

Reflexive: x+x= 2.x and not 3.x so its not reflexive
Symmetric: x+y = 3k so y+x = 3k
Transitive: xRy and yRz
x+y=3k for keZ
x+y=3n for neZ
So x=3k-y and y=3n-y so x+z=3k-y+3n-y= 3(k+n)-2y We see that its not transitive

Because it hasnt al three properties it isn't an equivalence relationship

I don't understand it really I read my textbook over and over but I still can't imagine how it works

BMath said:
Yes and this is my proof:

Reflexive: x+x= 2.x and not 3.x so its not reflexive
Even better would be to give a specific counterexample. Notice that if ##x = 3##, then ##x + x = 6 = 3\cdot 2## so the equation holds for some values of ##x##. Find one for which it does not hold.

Symmetric: x+y = 3k so y+x = 3k
OK. Writing in more detail, you might say something like "if ##xRy## then ##x + y = 3k##. Since addition of integers is commutative, we have ##x + y = y + x##. Therefore ##y + x = 3k##, which means that ##yRx##."

Transitive: xRy and yRz
x+y=3k for keZ
x+y=3n for neZ
I think you mean ##y + z = 3n## for that last equation. What can you conclude about ##x + z##?

From the definition you're given, xRy if and only if x + y = 3k, for some integer k. What this means is that for x and y to belong to the relation R, x and y have to add up to an integer multiple of 3. For example, 1R2 because 1 + 2 = 3 = 3 * 1.
BMath said:
Yes and this is my proof:

Reflexive: x+x= 2.x and not 3.x so its not reflexive
Some values of x belong to the relation, but others don't. As you note, x + x = 2x, so the only way a number x could belong to the relation is if x itself is a multiple of 3.
0R0 = 0 + 0 = 3 * 0
3R3 = 6 = 3 * 2
but 1 doesn't work, nor does 2.
BMath said:
Symmetric: x+y = 3k so y+x = 3k
Transitive: xRy and yRz
x+y=3k for keZ
x+y=3n for neZ
I think you mean y + z = 3n for n ##\in## Z.
BMath said:
So x=3k-y and y=3n-y so x+z=3k-y+3n-y= 3(k+n)-2y We see that its not transitive
If y is a multiple of 3, then we do have transitivity. For example, 0R3 because 0 + 3 = 3, and 3R6 because 3 + 6 = 9. From this, we see that 0R9 because 0 + 9 = 9 = 3*3.
BMath said:
Because it hasnt al three properties it isn't an equivalence relationship

jbunniii said:
Even better would be to give a specific counterexample. Notice that if ##x = 3##, then ##x + x = 6 = 3\cdot 2## so the equation holds for some values of ##x##. Find one for which it does not hold.

OK. Writing in more detail, you might say something like "if ##xRy## then ##x + y = 3k##. Since addition of integers is commutative, we have ##x + y = y + x##. Therefore ##y + x = 3k##, which means that ##yRx##."

I think you mean ##y + z = 3n## for that last equation. What can you conclude about ##x + z##?

That x+z is not an integer?

BMath said:
That x+z is not an integer?
No. The context of what both jbunniii and I said was the part where you were investigating transitivity or the relation. You're given that xRy and yRz, so do x and z also belong to the relation?

x,y,z are integers. xRy and yRz
0R2 and 2R5
0+2=2=3k and 2+5=7=3k we see that there doesn't exist an integer k such that 3k=2 and 3k=7 so
xRz doesn't exists.

BMath said:
x,y,z are integers. xRy and yRz
0R2 and 2R5
But 0 and 2 don't belong to the relation R, so you can't say 0R2. And 2 and 5 don't belong to the relation either, so you can't say 2R5.

Whatever numbers x, y, and z you work with, x and y have to belong to the relation, and y and z have to belong as well.
BMath said:
0+2=2=3k and 2+5=7=3k we see that there doesn't exist an integer k such that 3k=2 and 3k=7 so
xRz doesn't exists.

but then its always transitive cause i can't find any false relations 3R6 and 6R9
x+z=24=3*8

BMath said:
but then its always transitive cause i can't find any false relations
That's not a good reason. Possibly you didn't look hard enough.
BMath said:
3R6 and 6R9
x+z=24=3*8
How did you get 24? x = 3 and z = 9, so x + z = 12.

BTW, it's probably better to work with this algebraically rather than to try to guess combinations that work or don't work.

For the record, you have already shown enough to convince me or your instructor that R as defined in this thread is NOT a relation. It is not in general reflexive (e.g. 1 and 1 aren't in the relation, nor are 2 and 2, but 0 and 0 are, as are 3 and 3).

Also, the relation is transitive for some values of x, y, and z, but not for all such values. Most of what we've been doing here is trying to help you get a handle on what this is about.

BMath said:
but then its always transitive cause i can't find any false relations 3R6 and 6R9
x+z=24=3*8
How did you get 24? If x = 3, y = 6, and z = 9, then x + z = 12. [EDIT: as Mark44 already pointed out before I submitted my reply.]

Also, to prove transitivity, you must show that it is true in general, not for a specific example. Start by assuming that x + y and y + z are both multiples of 3. Then prove that this implies that x + z is also a multiple of 3. If you cannot show this in general, then try to find a counterexample.

Ah yes I see it
So if I have x+y 1R2 1+2=3=3.1
and y+z 2R7 2+7=9=3.3
then x+z 1R7 1+7=8=3.k we see that there's no integer k so that 3k=8 thus the relation is not transitive

So if its wrong I only have to give a counterexample? And if its true you need to prove it

Do you understand that this is saying that two numbers are "related" in this way if and only if their sum is a multiple of 3.
Yes, to prove that a general statement is NOT true you only have to give one example in which is does not work- a "counter-example".

2 is NOT related to itself in this way because 2+ 2= 4 is not a multiple of 3. Therefore, it is not "reflexive".

It is obviously symmetric.

4+ 2= 6= 2(3) so 4 and 2 are related. 2+ 1= 3 so 2 and 1 are related. But 4+ 1= 5 which is NOT a multiple of 3 so this is NOT transitive.

HallsofIvy said:
Do you understand that this is saying that two numbers are "related" in this way if and only if their sum is a multiple of 3.
Yes, to prove that a general statement is NOT true you only have to give one example in which is does not work- a "counter-example".

2 is NOT related to itself in this way because 2+ 2= 4 is not a multiple of 3. Therefore, it is not "reflexive".

It is obviously symmetric.

4+ 2= 6= 2(3) so 4 and 2 are related. 2+ 1= 3 so 2 and 1 are related. But 4+ 1= 5 which is NOT a multiple of 3 so this is NOT transitive.

Yes thank you very much for the help!

## 1. What is the purpose of "Proving Relation Properties: A Guide"?

The purpose of "Proving Relation Properties: A Guide" is to provide a comprehensive and step-by-step guide for scientists to prove relation properties in their research. This guide aims to help scientists understand and apply the principles of relation properties in a scientific context.

## 2. What are relation properties?

Relation properties refer to the characteristics or properties of a relationship between two or more variables. These properties can include symmetry, reflexivity, transitivity, and more.

## 3. Why is it important to prove relation properties?

Proving relation properties is important because it helps scientists to establish the validity and reliability of their research findings. It also allows for a deeper understanding of the relationships between variables and can help to identify any potential biases or limitations in the research.

## 4. What are some common techniques for proving relation properties?

Some common techniques for proving relation properties include using mathematical proofs, conducting experiments, and analyzing data using statistical methods. It is important for scientists to choose the most appropriate technique based on the specific relation properties being studied.

## 5. Can relation properties be proven in all scientific fields?

Yes, relation properties can be proven in all scientific fields where there is a relationship between variables. This includes fields such as biology, psychology, physics, and more. However, the specific techniques and methods used for proving relation properties may vary depending on the field of study.