My discrete book says that the set $A = \{4,5,6\}$ and the relation $R = \{(4,4),(5,5),(4,5),(5,4)\}$ is symmetric and transitive but not reflexive.
I was wondering how this is possible, because if a set $A$ is symmetric, doesn't it also need to include $(5,6),(6,5),(4,6),(6,4)$?
Also if it's transitive, doesn't it have to include $(4,5),(5,6),(4,6)$?
I thought the definition of a transitive relation was that $(x$ $R$ $y)$ $\land$ $(y$ $R$ $z)$ then $(x$ $R$ $z)$.
Best Answer
A set can't be symmetric; a relation can be. (By the way, it's possible for a set to be "transitive", but that doesn't mean the same thing as a transitive relation: a transitive set is a set $x$ such that if $z \in y$ and $y \in x$, then $z \in x$.)
A relation on a set need not involve every member of the set. For example, the relation on $\mathbb{N}$ given by "is a prime divisor of" doesn't touch $1$ at all: $1$ is not related to anything and nothing is related to it. In your example, $6$ is not related to anything by $R$, and nothing is related to $6$ by $R$.
"Symmetric" just means that if $a \sim b$, then $b \sim a$. Note that it doesn't tell us about any elements of $A$ we haven't seen before: from the mere knowledge that $4 \sim 5$, we can't use symmetry to deduce that anything is related to $6$. Similarly transitivity.