When do mathematical definitions matter?

In the movie Hanna, Hanna was a 16 year old  girl who grew in the wilderness of Finland. Since two years old, she never had contact with the real world and modern technology.  Hanna learned about the world from Erik, his adopted father, and by reading books.  In one of the father-daughter conversations, Hannah asked about music. Their conversation was as follows.

Erik: Music is a combination of sounds with a view of beauty to form.
Hanna:  But how does it feel?
Erik: Good. It feels good. It’s, uh,nice.
Hanna:  Tell me properly. Can you play music?
Erik: Your mother could. She used to sing very well.
Hanna:  I’d like to hear it for myself.

Hanna probably feels the way students feel when we define an unfamiliar concept.  To students who are mathematically matured, definitions at the beginning of the lesson are probably understood, but for those who are just learning the basic concepts, it might be a little vague. Imagine a teacher saying that a function is a correspondence between two sets A and B, where each element in A has exactly one and only one corresponding element in B to students who have no prior experience of the said relationship.

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