confusion on "anti-symmetric" and "symmetric"

Shing

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.

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 [itex]\subseteq[/itex] is anti-symmetric: if [itex]A\subseteq B[/itex] and [itex]B\subseteq A[/itex] then A=B.
The relation "=" is symmetric: if A=B then B=A.

Shing

thank you so much!
I can see the difference now.

would you mind elaborating please? :D

Landau

Elaborate on what?

yossell

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.

Shing

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")

JSuarez

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 [itex]\leq[/itex] 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.

SW VandeCarr

Informally: Your boss can fire you, but you can't fire your boss. You are not equal.

Shing

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)