Relation symmetric confusion

Question

So Symmetric = (a,b), (b,a)

Set = {<1, 1>, <1, 2>, <1, 4>, <2, 1>, <2, 2>, <3, 3>, <4,1 >, <4, 4>}

I understand <1,2> and <2,1> is symmetric, but are <2,2>, <3,3> and <4,4> necessary here to form a symmetric set?

Could someone please specify why is this set a symmetric?

Answer 1

You should write the definition more carefully. Symmetric means $(a,b) \in R \implies (b,a) \in R$ Note that this says the empty relation is symmetric. You can have $(2,2)$ (and the others) in or out without changing the fact that the relation is symmetric. The way to destroy symmetry would be to remove $(1,2)$ without removing $(2,1)$ or to add $(2,3)$ without adding $(3,2)$

Answer 2

Symmetry means that $$ (a,b)\in R\Rightarrow (b,a) \in R $$ Here, <2,2>, <3,3>, and <4,4> are not required to be in the set. If they were, then the set would be reflexive, which is defined as $$ R\subseteq A\times A\\ \forall a\in A \Rightarrow (a,a) \in R $$

Answer 3

The pairs you list in your set $R$ are not symmetric; it is the relation $R$ itself that is symmetric. Symmetry is a property of a relation and not a property of the elements that are related.

To be a symmetric relation, we must have that for all $x, y$ in the set on which the relation is defined, that if it's the case that $\langle x, y\rangle \in R$, then so must be $\langle y, x\rangle \in R$. If some pair $\langle a, b\rangle \notin R$, then we can't have $\langle b, a\rangle \in R$.

Another way to think of this is that a relation $R$ IS Symmetric...unless there exists a pair $\langle x, y\rangle \in R$, but $\langle y, x\rangle \notin R$.

In the case of $\langle x, x\rangle \in R$, it is trivially true that by symmetry, $\langle x, x\rangle \in R$. But it is not necessary that the pairs of the form $\langle x, x\rangle$ are in the relation $R$ to be symmetric. It would be symmetric without pairs of that form.