# Give a set and a relation that satisfies the properties

• MHB
• mathmari
In summary, the four properties are reflexive, symmetric, antisymmetric, and transitive. A relation that satisfies these properties is symmetric, antisymmetric, and transitive.
mathmari
Gold Member
MHB
Hey!

I am looking at the following:

There are the terms reflexive, symmetric, antisymmetric and transitive.
Give for each combination of the properties (if possible) a set $M$ and a relation $R$ on $M$, such that $R$ satisfies these properties. What is meant exactly? Every possible combination? So do we have to give a set and a relation that satisfies the below properties?
• reflexive, symmetric
• reflexive, antisymmetric
• reflexive, transitive
• symmetric, antisymmetric
• symmetric, transitive
• antisymmetric, transitive
• reflexive, symmetric, antisymmetric
• reflexive, symmetric, transitive
• reflexive, antisymmetric, transitive
• symmetric, antisymmetric, transitive
• reflexive, symmetric, antisymmetric transitive

(Wondering)

Well, you have four properties so 4!= 24 possible "combinations" (25 if you include the "empty combination"- that none of those properties are true).

I think you should give special consideration to two of those properties. What must be true of a set so that it will be both "symmetric" and "anti-symmetric"?

HallsofIvy said:
Well, you have four properties so 4!= 24 possible "combinations" (25 if you include the "empty combination"- that none of those properties are true).

Since the order of the properties doesn't matter do we not have $\binom{4}{2}+\binom{4}{3}+\binom{4}{4}=11$ combinations?
HallsofIvy said:
I think you should give special consideration to two of those properties. What must be true of a set so that it will be both "symmetric" and "anti-symmetric"?

A relation R is symmetric if it holds $aRb \iff bRa$ for $a,b\in M$.
A relation R is ant-symmetric if it holds $aRb$ then it doesn't hold that $bRa$ for $a\neq b\in M$.

So this combination is not possible, right? (Wondering)

HallsofIvy said:
Well, you have four properties so 4!= 24 possible "combinations" (25 if you include the "empty combination"- that none of those properties are true).
I believe there are $2^4=16$ possible combinations of 4 properties.

mathmari said:
A relation R is symmetric if it holds $aRb \iff bRa$ for $a,b\in M$.
A relation R is ant-symmetric if it holds $aRb$ then it doesn't hold that $bRa$ for $a\neq b\in M$.

So this combination is not possible, right?
It is possible.

Evgeny.Makarov said:
It is possible.

Ah yes! For the empty set, right?

So do we have the following?

We consider the set $M=\{0,1,2\}$.

• reflexive, symmetric : $R=\{(0,0), (1,1), (2,2), (0,1), (1,0)\}$.
• reflexive, anti-symmetric : $R=\{(0,0), (1,1), (2,2), (0,1)\}$.
• reflexive, transitive : $R=\{(0,0), (1,1), (2,2), (0,1), (1,2), (0,2)\}$.
• symmetric, anti-symmetric : $R=\emptyset$.
• symmetric, transitive : $R=\{(0,0), (0,1), (1,0)\}$.
• anti-symmetric, transitive : $R=\{(0,0), (1,1), (0,1)\}$.
• reflexive, symmetric, anti-symmetric : $R=\emptyset$.
• reflexive, symmetric, transitive : $R=\{(0,0), (1,1), (2,2), (0,1), (1,0), (1,2),(2,1),(0,2), (2,0)\}$.
• reflexive, anti-symmetric, transitive : $R=\{(0,0), (1,1), (2,2), (0,1), (1,2),(0,2), (2,1)\}$.
• symmetric, anti-symmetric, transitive : $R=\emptyset$.
• reflexive, symmetric, anti-symmetric, transitive : $R=\emptyset$.

(Wondering)

mathmari said:
For the empty set, right?
No, for any subset of the diagonal relation $\Delta=\{(x,x)\mid x\in M\}$.

I don't have time now to check all your answers, but as I said, there are 16 combinations, and they should be enumerated in a systematic way that makes it easy to see that nothing is missing. In your enumeration the first item "reflexive, symmetric" does not mean "reflexive, symmetric, not antisymmetric and not transitive" because this relation is transitive.

Evgeny.Makarov said:
No, for any subset of the diagonal relation $\Delta=\{(x,x)\mid x\in M\}$.

I don't have time now to check all your answers, but as I said, there are 16 combinations, and they should be enumerated in a systematic way that makes it easy to see that nothing is missing. In your enumeration the first item "reflexive, symmetric" does not mean "reflexive, symmetric, not antisymmetric and not transitive" because this relation is transitive.

So a relation that is symmetric, anti-symmetric and transitiveis is every relation in the form $\{(x,x)\mid x\in M\}$, right? But the empty set satisfies also these properties, or not? (Wondering)

mathmari said:
So a relation that is symmetric, anti-symmetric and transitiveis is every relation in the form $\{(x,x)\mid x\in M\}$, right?
$\{(x,x)\mid x\in M\}$ is a fixed set. You would not say, "Any number of the form 5", right? But yes, a relation is both symmetric and antisymmetric iff it is a subset of the diagonal relation $\Delta=\{(x,x)\mid x\in M\}$. The empty relation is also a subset of $\Delta$ and is symmetric and antisymmetric.

Evgeny.Makarov said:
$\{(x,x)\mid x\in M\}$ is a fixed set. You would not say, "Any number of the form 5", right? But yes, a relation is both symmetric and antisymmetric iff it is a subset of the diagonal relation $\Delta=\{(x,x)\mid x\in M\}$. The empty relation is also a subset of $\Delta$ and is symmetric and antisymmetric.

Ah ok!

So $\Delta$ is symmetric and antisymmetric and so also transitive, right? (Wondering)

Yes, every subset of $\Delta$, including $\Delta$ itself, is transitive.

Evgeny.Makarov said:
Yes, every subset of $\Delta$, including $\Delta$ itself, is transitive.

Ok, I got it! Thank you! (Smile)

## 1. What is a set in relation to mathematics?

A set in mathematics is a collection of distinct objects, called elements, that are grouped together based on a common characteristic or property. Sets are typically denoted using curly braces { } and the elements are separated by commas.

## 2. What is a relation in mathematics?

A relation in mathematics is a connection or association between two or more sets. It is a set of ordered pairs, where the first element in each pair belongs to the first set and the second element belongs to the second set. Relations can be represented using tables, graphs, or equations.

## 3. How can a set and a relation satisfy certain properties?

A set and a relation can satisfy certain properties if they adhere to specific rules or conditions. For example, a relation can be symmetric if for every ordered pair (a,b), there exists an ordered pair (b,a) in the relation. Similarly, a set can satisfy properties such as being finite or infinite, being empty or non-empty, or containing only certain types of elements.

## 4. Can you give an example of a set and a relation that satisfies certain properties?

Yes, for example, the set of even numbers {2, 4, 6, 8, ...} and the relation "is a multiple of" satisfies the property of being reflexive, as every even number is a multiple of itself. It also satisfies the property of being symmetric, as if a is a multiple of b, then b is also a multiple of a.

## 5. How are sets and relations used in real-life applications?

Sets and relations are used in various real-life applications, such as in database management, social network analysis, and computer algorithms. For example, in a social network, people can be represented as elements in a set, and the relationships between them can be represented as relations. This can be used to analyze patterns and connections within a social network.

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