How come the relation $\subseteq $ on the power set $2^N$ is antisymmetric?

Question

where $2^N$ is the power set with $n$ elements (subsets).

Does it hold true to any set or just the power set $2^N$?

Answer 1

By definition, two sets $A$ and $B$ are equal if and only if $A\subseteq B$ and $B\subseteq A$. Now a relation $R$ on a set $X$ is antisymmetric if $aRb$ and $bRa$ implies $a=b$ for all $a,b\in X$. Does this help you see why $\subseteq$ is antisymmetric?

Answer 2

Let me give an intuitive, not pure explanation. First, start out by understanding that antisymmetic means that for $x$ and $y$ of a set, $xRy$ implies $yRx$ is not the case, e.g., a bunch of males and none are brothers. So if we look at symmetric relations, i.e., symmetry is when $xRy$ does imply $yRx$, e.g., two different males, $x$ and $y$ satisfy the "is the brother of" relation. But again, symmetry assumes $x$ and $y$ are different, otherwise it's just reflexive, e.g., somebody is their own brother. In a way anti-symmetric is taking away the possibility that $x$ and $y$ are both different and symmetric, e.g., Josh and Will being brothers would be symmetric -- they're different guys and they're each other's brother. So insisting on the implication that Josh and Will are brothers, $xRy$ and $yRx$, only if they are the same person (one name is an alias?) flips the situation into antisymmetry. That is, when we insist $x$ and $y$ have to be the same element to be in a relation, we've taken away the possibility that any two elements of a relation can exhibit symmetry. So yes, we're still working in the context, language of symmetry, i.e., $xRy$, $yRx$, but creating exactly its opposite by insisting $x$ and $y$ must be the same thing (aliases) for $xRy$ and $yRx$ to hold; otherwise they're all in antisymmetric relationships. Long short, this is all just a round-about way of saying that if $xRy$ is true, then automatically $yRx$ is not true -- but doing it in a reductio ad absurdum way, I guess. Another example would be a set of natural numbers. Here "less than or equal to" is an antisymmetic relation on the set, and only when two different elements are actually the same are they also equal. I'm guessing all of this is necessary because math needs to have a precise way of giving the opposite of symmetry and not have it confused with asymmetry. Again, YMMV.