Difference between == and === in Mathematica

Question

I was under the impression that = is an assignment, == is a numeric comparison, and === is a symbolic comparison (as well as in some other languages == being equal to and === being identical to. However, looking at the following it would appear that this is not necessarily the case...

In: x == x
Out: True

In: x === x
Out: True

In: 5 == 5
Out: True

In: 5 === 5
Out: True

In: x = 5
Out: 5

In: 5 == x
Out: True

In: 5 === x
Out: True

In: 5 5 == 5x
Out: True

In: 5 5 === 5x
Out: True

In: x == y
Out: x == y

In: x === y
Out: False

In: y = x
Out: 5

In: x == y
Out: True

In: x === y
Out: True

So what exactly is the difference between == and === in Mathematica? I have been looking at the documentation but I still don't quite understand it.

Answer 1

Equal refers to semantic equality whereas SameQ is syntactic equality. For instance, Sin[x]^2+Cos[x]^2 and 1 are the same number, so they are equal semantically. Since this can not be determined without more transformations, Equal returns unevaluated. However, actual expressions are different, so SameQ gives False.

Sin[x]^2 + Cos[x]^2 == 1
Sin[x]^2 + Cos[x]^2 === 1
Simplify[Sin[x]^2 + Cos[x]^2 == 1]

Note that there's special handling of Real numbers, SameQ[a,b] can return True if a and b differ in the last binary digit. To do more restrictive identity testing, use Order[a,b]==0

a = 1. + 2^-52;
b = 1.;
a === b
Order[a, b]==0

SameQ can return True for expressions that are syntactically different because expression heads may sort arguments automatically. You can prevent automatic sorting by using holding attributes. For instance

c + d === d + c
SetAttributes[SameQ, HoldAll]
c + d === d + c