Equal vs Equivalent: Finer differences in meaning and usage? in 4 distinct scenarios outlined?

"Equal" and "equivalent" are equivalent, but they're not equal. :-)

They have similar, but not identical meanings. Equal means the same thing, but equivalent means that one can frequently be substituted for the other.

They have different meanings in mathematics.  As Mike says, equality (represented by “=”) means that two things are the same.  For example, 3 = 3 = 3.0 = 1+2 … and an infinite variety of other ways of expressing the same number, but 3 does not equal any other number.

Equivalence is much more loosely defined.  An equivalence relationship (typically represented by “≅”) is any relationship that satisfies the following properties:

• Reflexive: For any object X, it is true that X ≅ X.
• Symmetric: For any X and Y, if X ≅ Y, then Y ≅ X.
• Transitive: For any X, Y, and Z, if X ≅ Y and Y ≅ Z, then X ≅ Z.

Clearly equality satisfies the above properties, so equality is an equivalence relationship.  (In other words, equality is a subset (a special case) of equivalence.)  But equivalence relationships can be more interesting.  One of the best known equivalence relationships is modulo.  For example, in the modulo 10 equivalence relationship, 3 ≅ 13 ≅ 23 ≅ 33 …