Proving properties of an equivalence relation

Question

For f: A $\rightarrow$ A , I need to show that a~b if a=b is an equivalence relaion.

I draw a mapping that looks like:

a b c

a a,a a,b a,c

b b,a b,b b,c

c c,a c,b c,c

1. Reflexivity holds a~a if a=a. So I look in my mapping where a=a. I choose (a,a) from my mapping because it feels right but I don't understand how this implies a=a. How is it that (a,a) imply a=a?

2. Symmetry holds a~b $\Rightarrow$b~a if a=b $\Rightarrow$ b=a. I choose (a,b) and (b,a) from my mapping again because it feels good but I don't understand how this implies that a=b and b=a. How is it that these two pair have a symmetric relation?

3. Transitivity holds a-b,b-c $\Rightarrow$ a~c. I choose (a,b), (b,c), and (c,d) from my mapping but I don't understand how this implies that a=b, b=c, and a=c. How is it that these three pair have a transitive relation?