I) Take an English-into-English dictionary (any other language will also do). Start with any word and note down any word occurring in its definition, as given in the dictionary. Take this new word and note down any word appearing in it until a vicious circle results. Prove that a vicious circle is unavoidable no matter which word one starts with , (Caution: the vicious circle may not always involve the original word).
For example, in geometry the word “point” is undefined. For example, in set theory, when we write or say : ; the element “a” ‘belongs to’ “set A” —- the word “belong to” is not defined.
So, in all branches of math or physics especially, there are such “atomic” or “undefined” terms that one starts with.
After such terms come the “axioms” — statements which are assumed to be true; that is, statements whose proof is not sought.
The following are the axioms based on which equations are solved in algebra:
- If to equals we add equals, we get equals.
- If from equals we take equals, the remainders are equal.
- If equals are multiplied by equals, the products are equal.
- If equals are divided by equals (not zero), the quotients are equal.
Discuss the following “proof” of the (false) theorem:
If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:
PROOF BY INDUCTION:
If , the result is evident.
Step 2: By the induction hypothesis the result is true when ; we must prove that it is correct when . Let S be any set containing exactly real numbers and denote these real numbers by . If we omit from this list, we obtain exactly k numbers ; by induction hypothesis these numbers are all equal:
If we omit from the list of numbers in S, we again obtain exactly k numbers ; by the induction hypothesis these numbers are all equal:
It follows easily that all numbers in S are equal.
Comments, observations are welcome 🙂
(shared from Clay Math website and shared for my readers. Many thanks to James Maynard and Clay Math :-))
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 !!
1) Solve in real numbers the system of equations:
Hints: do you see some quadratics ? Can we reduce the number of variables? …Try such thinking on your own…
2) Let be real numbers such that and . Prove that .
3) Let a, b, c be positive real numbers. Prove that