Mathematic Definitions and Ideas

[Query]

TL;DR Summary

I have never posted before. But, I had a question:

What is the mathematical process of "opposite," using the general definition of the word?

What is opposite in Mathematics?

I argued with my engineering brother and his pal that "inverse" is indeed "opposite" if you use the general concept of "opposite." My brother claims that the idea of opposite in mathematics only means returning to null.

Opposite: 1.) Having a position on the other side. 2.) Diametrically different.

Am I totally wrong? Or is the idea of opposite not very applicable?

 

[PeroK]

Summary: I have never posted before. But, I had a question:

What is the mathematical process of "opposite," using the general definition of the word?

What is opposite in Mathematics?

I argued with my engineering brother and his pal that "inverse" is indeed "opposite" if you use the general concept of "opposite." My brother claims that the idea of opposite in mathematics only means returning to null.

Opposite: 1.) Having a position on the other side. 2.) Diametrically different.

Am I totally wrong? Or is the idea of opposite not very applicable?

"Opposite" numbers mean numbers that are the negative of each other. Apart from that, I'm not aware of "opposite" having any strict mathematical definition.

Instead, there are terms like "converse", "inverse" and "negation", which are all well-defined.

 

[fresh_42]

There is no specific opposite in mathematics. Depending on context it can refer to:

• inversion
• subtraction
• complement
• dualism
• anti isomorphisms
• indirect conclusion
• contraposition

The only occasion I can remember to have actually seen something like $G^{opp}$ was when a binary operation of $G$ has been turned from left to right to right to left. So the list goes on with

• reflexion
• conversion
• the two parts of an equivalence relation or conclusion

 

[nuuskur]

When talking about a proposition one could say "well, consider the opposite.." i.e consider the negation of said proposition. We don't really have anyone specific instance where we say "opposite" about something. In view of @fresh_42 's examples, I prefer to call all those operations what they are instead of potentially confusing the reader/listener - unnecessary.