Confusion on anti-symmetric and symmetric

[Shing]

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.

 

[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.

 

[Shing]

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

 

[Landau]

Elaborate on what?

 

[yossell]

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.

 

[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 $\leq$ 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.

 

[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)