# Wisdom of Abraham A. Fraenkel

Before looking at a proof, the student should realize what is to be proved, why a proof is necessary, and what is the main difficulty to be overcome.

# Difference between identity and equality

A nice way to illustrate this is the following conversation in Lewis Caroll’s Alice in Wonderland:

In a discussion between Alice and the Pigeon, the latter accuses her of being a serpent. In spite of her indignation, Alice has to confess that “little girls eat as many eggs as serpents do.” But, the Pigeon calls “anything with long necks and eating many eggs” as “a serpent.”

In other words, two objects may have same property and are equal in that sense. But, the objects might be different.

So, that brings to mind the next question: what is the difference between equal and equivalence. In my opinion, equivalence measures equality up to a certain property. For example, congruence modulo relation in numbers. Or, congruent triangles in Euclid’s geometry.

Regards,

Nalin Pithwa