TITLE: Antisymmetric Relations
QUESTION [27 upvotes]: Given a set $\{1,2,3,4\}$, how is the following relation $R$ antisymmetric?
$$R = \{(1, 2), (2, 3), (3, 4)\}$$
Note: Antisymmetric is the idea that if $(a,b)$ is in $R$ and $(b,a)$ is in $R$, then $a = b$. In my textbook  it says the above is antisymmetric which isn't the case as whenever $(a,b)$ is in $R$, $(b,a)$ is not.

REPLY [5 votes]: In my opinion your misunderstanding is in the logic, not in the set theory.
If $p$ is false, then the conditional statement “if $p$ then $q$” is vacuously true. This does not mean that $q$ is true, it means the entire statement is true.
As @Doug Chatham says,

For example, consider the implication, "If 2+2=5, then you will pass the course." Since 2+2 is not 5, the statement is a true statement, regardless of whether or not you pass the course.

Regarding your question, if $(a,b)\in R$ and $(b,a)\in R$ together imply $a=b$, and either one (or both) of $(a,b)\in R$ or $(b,a)\in R$ is false, then the entire statement is true: R is antisymmetric.
The definition of antisymmetry does not state

if $(a,b) \in R$ then $(b,a) \in R$

(that would be the definition of symmetry).