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Why is a symmetric relation defined by $forall{x}forall{y}(xRy implies yRx)$ and not $forall{x}forall{y}(xRy iff yRx)$?
(I have only found a couple of sources that defines it with a biconditional)

For example, according to Wolfram:

A relation $R$ on a set $S$ is symmetric provided that for every $x$ and $y$ in $S$ we have $xRy iff yRx$.

But the majority of books defines it the other way.
And I think I agree with the second definition.

Because if we use the first definition with "$implies$", we know the truth table of the implication in particular $P implies Q$ is true when $P$ is false and $Q$ is true. That means in the context of symmetric relation that $(x,y) notin R implies (y,x) in R$ is true.

And the example $A = {1,2,3,4}$ with relation $R = {(2,1),(3,1),(4,1)}$ satisfies the definition because $(x,y) notin R implies (y,x) in R$ is true.

And for me it's weird that this case is considered symmetric.
Or maybe I have a profound confusion with the concept.
I would like that you guys help me clarify. *Sorry for my grammar I'm not a native english speaker.

For all $x$ and all $y$ make the if and only if unnecessary (albeit perfectly acceptable).

1) $(x,y) in R implies (y,x) in R$ for ALL $x,y in A$

And the statement 2) $(x,y) in R iff (y,x) in R$ are equivalent statements.

If 1) is true and $(x,y) not in R$ then although $(x,y)in Rimplies (y,x)in R$ or $F implies (y,x)in R$ is true, it does not tell us any thing about whether or not $(y,x) in R$. However $(y,x) in R implies (x,y) in Y$ tells us that $(y,x) not in R$. Because $(y,x) in R implies (x,y) in R$ means $(y,x) in R implies F$. An the only thing that implies a false statement is a false statement. So we must have $(y,x) not in R$.

So in your example you have $(1,2)in Rimplies (2,1)in R$ is true but you don't have $(2,1) in R implies (1,2) in R$ as true.

So it isn't symmetric.

=====

Another way to look at it:

If $A = {1,2,3}$

Then we will have 9 statments.

By 1) the nine statements are:

$(1,1)in Rimplies (1,1) in R$

$(1,2) in R implies (2,1) in R$

$(1,3) in R implies (3,1) in R$

$(2,1) in R implies (1,2) in R$

... etc... all nine are needed.

With 2) we also have nine statements:

$(1,1)in Riff (1,1) in R$

$(1,2) in R iff (2,1) in R$

$(1,3) in R iff (3,1) in R$

$(2,1) in R iff (1,2) in R$

...etc....

$(1,2) in R iff (2,1) in R$ and $(2,1) in R iff (1,2)in R$ is redundant.

So aesthetically, using definition 2) is .... inefficient.

If $A$ is a set and $R$ is a binary relation defined on $A$ (that is, $R$ is a subset of $Atimes A$), then the usual definition of symmetry (as far as $R$ is concerned) is$$(forall xin A)(forall yin A):xmathrel Ryimplies ymathrel Rx.tag1$$And this is equivalent to$$(forall xin A)(forall yin A):xmathrel Ryiff ymathrel Rx.tag2$$So, why do we choose $(1)$ instead of $(2)$ in general? Because, in general (although not in this case) it is easier to verify the assertion $Aimplies B$ than $Aiff B$. And (again, in general), when we choose between two distinct but equivalent definitions, we usually choose the one which is easier to verify that it holds.

If $xRy implies yRx$ for all $x$ and all $y$, then we can choose $x := tilde{y}$ and $y := tilde{x}$ and get $tilde{y}Rtilde{x} implies tilde{x}Rtilde{y}$ or, equivalently, $yRx implies xRy$, which is $impliedby$.

In conclusion, since the implication should hold for all $x,y$, the equivalence already holds.

Let it be that $R$ is a symmetric relation.

This according to the first mentioned definition:$$forall xforall y[xRyimplies yRx]tag1$$

Now let it be that $aRb$.

Then we are allowed to conclude that $bRa$.

On the other hand if $bRa$ then also we are conclude that $aRb$.

So apparantly we have:$$aRbiff bRa$$

Proved is now that for a symmetric relation $R$ (based on definition $(1)$) we have:$$forall xforall y[xRyiff yRx]tag2$$

So $(2)$ is a necessary condition for $(1)$.

Next to that it is obvious that $(2)$ is also a sufficient condition for $(1)$ so actually the statements are equivalent.

Both can be used as definition then, but in cases that like that it is good custom to go for the one with less implications.