Why is a Symmetric Relation also Transitive?

Question

A relation R on set A is as follows:

R = {(1,1), (2,2), (3,3)}

R is symmetric!

But WHY is R Transitive?

Answer 1

Many symmetric relations are not transitive; for example: A lives within one mile of B. So your title would make more sense if instead of "Why is a symmetric relation transitive?" it said "Why is this symmetric relation transitive?".

It is transitive because it lacks any opportunity not to be transitive: you would need to find $a, b, c$ such that $a\sim b$ and $b\sim c$ but $a\not\sim c$ in order to have a relation that is not transitive. The only way you can say $a\sim b$ and $b\sim c$ with this relation is if $a=b=c$. And in that case $a\sim c$ holds.

Answer 2

All you need for transitivity is $$(a,b)\in R, (b,c) \in R \Rightarrow (a,c)\in R$$ But $(a,b)\in R$ implies $a=b$ (why?), so $a=b=c$ and thus transitivity follows immediately from reflexivity.

Answer 3

To prove it, we are wanted to suppose that $(a,b), (b,c) \in R$. Our goal is to show that $(a,c) \in R$. However, For any different element $a, b, c \in A$. So $(a,b), (b,c) \in R$ cannot hold, and this shows the transitive is true.