What is the difference between equality and equivalence?

Question

I\'ve read a few instances in reading mathematics and computer science that use the equivalence symbol ≡, (basically an \'=\' with three lines)

Answer 1

The answers above are right / partially right but they don't explain what the difference is exactly. In theoretical computer science (and probably in other branches of maths) it has to do with quantification over free variables of the logical equation (that is when we use the two notations at once).

For me the best ways to understand the difference is:

1. By definition
   A ≡ B
   means
   For all possible values of free variables in A and B, A = B

   or

   A ≡ B <=> [A = B]

2. By example
   x=2x
   iff (in fact iff is the same as ≡)
   x=0

   x ≡ 2x
   iff (because it is not the case that x = 2x for all possible values of x)
   False

I hope it helps

Edit:

Another thing that came to my head is the definitions of the two.

A = B is defined as A <= B and A >= B, where <= (smaller equal, not implies) can be any ordering relation

A ≡ B is defined as A <=> B (iff, if and only if, implies both sides), worth noting that implication is also an ordering relation and so it is possible (but less precise and often confusing) to use = instead of ≡.

I guess the conclusion is that when you see =, then you have to figure out the authors intention based on the context.