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There are four elementary arithmetic operators. Are all operations in mathematics derived from the four elementary arithmetic operators?

I'm studying linear algebra and noticed that some exercises define a new operation such as "adition between (a,b) and (c,d) is (a+d, c+b)". Is there a freedom in mathematics to define new operations at wild?

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    $\begingroup$ Yes, yes! : ) $\,$ $\endgroup$
    – Berci
    Nov 7 '14 at 1:26
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    $\begingroup$ most Definitely! Define as many new operations as you want! $\endgroup$
    – Joao
    Nov 7 '14 at 1:28
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    $\begingroup$ There are many contest problems that start with something like "Let $a\#b=a^2-b$…" or something like that. $\endgroup$ Nov 7 '14 at 1:36
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I'm not an expert in this field, although I don't think all of the operations in the integers can be expressed in terms of those.

For example consider the operation $a\star b$ which is defined to be the $a$'th prime larger than $b$.

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