At the outset, let me put a little sweetener also: All I want to do is draw attention to the importance of symbolic manipulation. If you can solve this tutorial easily or with only a little bit of help, I would strongly feel that you can make a good career in math or applied math or mathematical sciences or engineering.
On the other hand, this tutorial can be useful as a “miscellaneous or logical type of problems” for the ensuing RMO 2019.
I) Let S be a set having an operation * which assigns an element a*b of S for any . Let us assume that the following two rules hold:
i) If a, b are any objects in S, then
ii) If a, b are any objects in S, then
Show that S can have at most one object.
II) Let S be the set of all integers. For a, b in S define * by a*b=a-b. Verify the following:
a) unless .
b) in general. Under what conditions on a, b, c is ?
c) The integer 0 has the property that for every a in S.
d) For a in S,
III) Let S consist of two objects and . We define the operation * on S by subjecting and to the following condittions:
Verify by explicit calculation that if a, b, c are any elements of S (that is, a, b and c can be any of or ) then:
iv) There is a particular a in S such that for all b in S
v) Given , then , where a is the particular element in (iv) above.
This will be your own self-appraisal !!