## confusion on "anti-symmetric" and "symmetric"

Hi guys,
I am a physics sophomore at next term, recently I am doing a reading on Naive Set Theory on my own. However, I got a few confusion.

The books said that if A is a subset of B and B is a subset of A, then A=B, but this set inclusion is anti-symmetric,

on the other hand, based on the axiom of extension. Two sets are equal iff they have the same elements, then. if A=B, then it is symmetric.

My question is:

a.) What meant by being symmetric & anti-symmetric
b.) what is the difference between the two approaches to "A=B"?

Thank you so much for reading :D
have a good day.
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
 Recognitions: Science Advisor The terms 'symmetric' and 'anti-symmetric' apply to a binary relation R: R symmetric means: if aRb then bRa. R anti-symmetric means: if aRb and bRa, then a=b. Thus the relation $\subseteq$ is anti-symmetric: if $A\subseteq B$ and $B\subseteq A$ then A=B. The relation "=" is symmetric: if A=B then B=A.

 Quote by Landau The terms 'symmetric' and 'anti-symmetric' apply to a binary relation R: R symmetric means: if aRb then bRa. R anti-symmetric means: if aRb and bRa, then a=b. Thus the relation $\subseteq$ is anti-symmetric: if $A\subseteq B$ and $B\subseteq A$ then A=B. The relation "=" is symmetric: if A=B then B=A.
thank you so much!
I can see the difference now.

would you mind elaborating please? :D

Recognitions:

## confusion on "anti-symmetric" and "symmetric"

Elaborate on what?

 Quote by Shing Hi guys, The books said that if A is a subset of B and B is a subset of A, then A=B, but this set inclusion is anti-symmetric, on the other hand, based on the axiom of extension. Two sets are equal iff they have the same elements, then. if A=B, then it is symmetric. My question is: a.) What meant by being symmetric & anti-symmetric b.) what is the difference between the two approaches to "A=B"? Thank you so much for reading :D have a good day.
There is no `other hand' and there are not two approaches to A = B. The axiom of extension is the key principle for set identity: two sets are identical iff they have the same members.

Now, if it is assumed that A is a subset of B and B is a subset of A, we can prove that they are identical, using this principle.

For if A is a subset of B then every member of A is a member of B. And if B is a subset of A, then every member of B is a member of A. Thus x is a member of A if and only if it is a member B. Thus A and B have the same members. Thus, by our principle, they are the same set.
 Thanks for answering, I start to understand it. but I am still confused by what practical difference between symmetric and anti-symmetric is? in this case(set), they produce same result to me (except the "path")
 In a symmetric relation, if a is related to b, then b must also be related to a (as happens, for example, in equality). If the relation is antisymmetric, then if a and b are both related to each other, they must be identical (as is the $\leq$ relation). In fact, antisymmetrical relations usually express some kind of weak ordering. Picture as a directed graph: in a symmetric relation, if there is an arc between two distinct nodes, then there must be another arc in the opposite direction; for antisymmetry, this can only happen if the nodes are identical.
 Informally: Your boss can fire you, but you can't fire your boss. You are not equal.
 so is picking up boys pick up girls, girls never pick up guys. therefore, it is no equality of male and female over the anti-symmetric relation "picking up" right? lol (well, that's true at least in Asia lol)