Why do we even need transitivity?

Consider the following relation: $x$ opposes $y$ if and only if $x = -y$.

Can you see that the "opposes" relation is symmetrical but not transitive?


Some relations are symmetric without being transitive.

For example if John and Jill are friends and Jill and Jeff are friends, it does not imply that John and Jeff are friends.

So friendship is a relation which is symmetric but not transitive.

Mathematically speaking, in geometry perpendicularity is symmetric but not transitive.

We like the relation $x=y$ be both symmetric and transitive.


It's easy to write down an example of a symmetric relation which fails to satisfy transitivity: on the set $\{a,b,c\}$ we say that $a \sim b$ and $b \sim a$ and $b \sim c$ and $c \sim b$ (I'm using a different symbol $\sim$ so as not to confuse the issue). So no, the symmetry axiom does not imply the transitivity axiom.

Perhaps you might want to think of the transitive law as it is stated in some translations of Euclid: thing which are equal to the same thing are equal to each other.

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