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For each of the eight subsets of {reflexive, symmetric, transitive}, find a relation on $\{1, 2, 3\}$ that has the properties in that subset, but not the properties that are not in the subset.

What I have done

This is the same question asked here.

I have found the same possible subsets. But for each of them instead of establishing the relationship via ordered pairs, I have defined them with an operation. The possible subsets of {reflexive, symmetric, transitive} are $T_0=\{\varnothing \}, T_1=\{\text{Reflexive}\}, T_2=\{\text{Symmetric}\}, T_3=\{\text{Transitive}\}, T_4=\{\text{Reflexive, Symmetric}\}, T_5=\{\text{Reflexive, Transitive}\}, T_6=\{\text{Symmetric, Transitive}\}, T_7=\{\text{Reflexive, Symmetric, Transitive}\}$.

For $T_{0}$, define the relation $\rho$ on $\{1, 2, 3\}$ via $a\rho b$ iff $a-b=1$.

For $T_{1}$?

For $T_{2}$, define the relation $\rho$ on $\{1, 2, 3\}$ via $a\rho b$ iff $ab$ is even.

For $T_{3}$, define the relation $\rho$ on $\{1, 2, 3\}$ via $a\rho b$ iff $a<b$.

For $T_{4}$, define the relation $\rho$ on $\{1, 2, 3\}$ via $a\rho b$ iff $|a-b|\leq 1$.

For $T_{5}$, define the relation $\rho$ on $\{1, 2, 3\}$ via $a\rho b$ iff $a\leq b$.

For $T_{6}$?

For $T_{7}$, define the relation $\rho$ on $\{1, 2, 3\}$ via $a\rho b$ iff $a= b$.

For $T_1$ and for $T_6$ it is easy to find the relation via ordered pairs, but defining the relation by means of an operation has not been possible for me, could you please help me? If I completed the list using ordered pairs, I think it would be inelegant.

Martin Sleziak
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Hopmaths
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1 Answers1

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For $T_1$ you could use $0\le a-b \le 1$.

No symmetry: We have $(3,2)$ but not $(2,3)$

No Transitivity: We have $(3,2)$ and $(2,1)$ but not $(3,1)$

For $T_6$ you could use $\min\{a,b\} \neq 1$. Which is just saying that neither $a$ or $b$ is 1.

Mason
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