Question

What is the difference between equality (=) and equivalence (≡)? What is the difference between equality (=) and equivalence (≡)? @Mathematics

Answer 1

In mathematics there is a type of binary relationship on a set called equivalence; you can think of it as a generalized form of equality where certain essential properties of equality hold, but not the particular properties.

If you've seen an equivalence relationship, then you'll have an example.

Answer 2

The three 'essential properties' of an equivalence relationship ~ on a set X, its axioms, is that
1. x ~ x for all x in X
2. x ~ y => y ~ x for all x,y in X
3. x ~ y and y ~ z => x ~ z for all x,y,z in X.

It's clear that on the real numbers the binary relation = is an example of an equivalence relation.

Answer 3

Okay. That makes sense. What would be an example of something where an equivalence relationship would be appropriate where an equality would not?

Answer 4

...but on the other hand there are equivalence relationships that are not equality.

For instance, on the integers, define ~ by
x~y iff x-y is divisible by 3.

With a bit of work you can see that ~ is an equivalence relationship, but it's clearly not equality, =.

Answer 5

Just to clarify, yakeyglee,
≡ doesn't mean equivalence.

~ meaning equivalence, is as JamesJ described, more is here on wikipedia: http://en.wikipedia.org/wiki/Equivalence_relation

≡ means "is defined as" or "is equal by definition to"... in general use. (some older papers use it as congruence).

Here's a handy page you can use to look up the math shorthand, which comes in handy :)
http://en.wikipedia.org/wiki/Table_of_mathematical_symbols