Reflexive, symmetric, transitive, and antisymmetric

For any set $A$, there exists only one relation which is both reflexive, symmetric and assymetric, and that is the relation $R=\{(a,a)| a\in A\}$.

You can easily see that any reflexive relation must include all elements of $R$, and that any relation that is symmetric and antisymmetric cannot include any pair $(a,b)$ where $a\neq b$. So already, $R$ is your only candidate for a reflexive, symmetric, transitive and antisymmetric relation.

Since $R$ is also transitive, we conclude that $R$ is the only reflexive, symmetric, transitive and antisymmetric relation.


Your answer is correct and you can easily generalize it to a set with more elements

Apparently the only solution to your question is the diagonal relation, $$R=\{(x,x)|x\in A \}$$ for any set A.

Tags:

Symmetric Functions

Relations

Related

Optimization explained to a middle school kid How to form a mental image of $\mathbf {ijk}=-1$ in quaternions? Why is $\sum_{i=1}^n a$ always irrational if $n>0$ and $a$ is irrational? Query on a Solution to the Problem: $\gcd(5a+2,7a+3)=1$ for all integer $a$. Geometric meaning of the quantity $|a|^2 |b|^2 |c|^2 - (a \cdot b)(b \cdot c)(c \cdot a)$ for non-coplanar vectors $a$, $b$, $c$ Why isn't this approach in solving $x^2+x+1=0$ valid? How many $5$-digit numbers with distinct digits are divisible by 3 (and generalize) How to easily create a polynomial function that gives a desired output? Is the set $\mathcal{A}$ of all matrices whose trace is $0$ nowhere dense in $\mathbb{M}_n(\mathbb{R}),n \ge 2$? Trying to arrange sums in neat way Verify a Matrix Proof Given $A = A^2$ "categorical" proof of a seemingly symmetric statement about Noetherian/Artinian modules

Recent Posts