If a relation is not symmetric shall we say that it is anti symmetric?

Question

If a relation is not symmetric shall we say that it is anti symmetric? In a quiz show's preliminary round i had this question looks easy but i couldn't answer.can anyone answer?

Answer 1

Symmetric and anti-symmetric relations are not complements of each other. Consider $\{(a,b),(b,a),(a,c)\}$ over $\{a,b,c\}$. It is neither symmetric nor anti-symmetric.

Answer 2

No. Anti-symmetry means that if $R(x,y)$ and $R(y,x)$ then $x=y$. In other words, for anti-symmetry you cannot have both $R(a,b)$ and $R(b,a)$ for different $a$ and $b$. But that is not the negation of symmetry.

For example, for $R$ to be not anti-symmetric, we could have both $R(a,b)$ and $R(b,a)$, but to then also make it not symmetric, we simply add $R(a,c)$, which is exactly what @Levent did in their answer.

Notice that this example is also not asymmetric (indeed, asymmetry implies anti-symmetry, and so if it is not anti-symmetric, then it is automatically not asymmetric either) so asymmetry is also not the negation of symmetry.

I suppose you could use a phrase like 'non-symmetry', but it's best just to say: it's 'not symmetric'. Likewise the example given is also 'not anti-symmetric' and 'not asymmetric'.

Maybe this helps: 'symmetry', 'asymmetry', and 'anti-symmetry' are all properties of 'nicely behaving' relations, so you can have relations that don't have any of these 'nice' features.

Answer 3

You need to note about the logic (the "logic" tag should be added).

A relation $R$ over the set $X$ is symmetric if and only if this statement is true: "For all $a$ and $b$ in $X$, $a$ is related to $b$ $\Leftrightarrow$ $b$ is related to $a$."

A relation $R$ over the set $X$ is antisymmetric if and only if there are no pair of distinct elements $a$ and $b$ (at all) satisfies $a$ is related to $b$ $\Leftrightarrow$ $b$ is related to $a$.

If a relation $R$ over the set $X$ has four elements, $a_{1},a_{2},b_{1},b_2$ that satisfies $a_1$ and $b_1$ are related to each other, but $a_{2}$ and $b_2$ are not related to each other, we can conclude that $R$ is not symmetric, but it also violates the condition for the antisymmetric or $R$ is neither of them.