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.
(shared from Clay Math website and shared for my readers. Many thanks to James Maynard and Clay Math :-))
I) Prove that every function can be represented as a sum of an even function and an odd function.
II)Let A, B, C be subsets of a set S. Prove the following statements and illustrate them with Venn Diagrams:
2a) The famous DeMorgan’s laws in their basic forms: and . Assume that both sets A and B are subsets of Set S. In words, the first is: union of complements is the complement of intersection; the second is: intersection of two complements is the complement of the union of the two sets.
Let us say that we need to prove: .
Proof: It must be shown that the two sets have the same elements; in other words, that each element of the set on LHS is an element of the set on RHS and vice-versa.
If , then and . This means that , and and . Since and , hence . Hence, .
Conversely, if , then and . Therefore, and . Thus, and , so that . QED.
III) Prove that if I and S are sets and if for each , we have , then .
It must be shown that each element of the set on the LHS is an element of the set on RHS, and vice-versa.
If , then and . Therefore, , for at least one . Thus, , so that .
Conversely, if , then for some , we have . Thus, and . Since , we have . Therefore, . QED.
IV) If A, B and C are sets, show that :
V) Let I be a nonempty set and for each let be a set. Prove that
5a) for any set B, we have :
5b) if each is a subset of a given set S, then
VI) Prove that if , , and are functions, then :
VII) Let be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:
7i) ; in words, image of union of two sets is the union of two images;
7ii) ; in words, image of intersection of two sets is a subset of the intersection of the two images;
7iii) ; in words, the inverse image of the union of two sets is the union of the images of the two sets.
7iv) ; in words, the inverse image of intersection of two sets is intersection of the two inverse images.
7v) ; in words, the inverse of the image of a set contains the set itself.
7vi) ; in words, the image of an inverse image of a set is a subset of that set.
For questions 8 and 9, we can assume that the function f is and a set A lies in domain X and a set C lies in co-domain Y.
8) Prove that a function f is 1-1 if and only if for all ; in words, a function sends different inputs to different outputs iff a set in its domain is the same as the inverse of the image of that set itself.
9) Prove that a function f is onto if and only if for all ; in words, the image of a domain is equal to whole co-domain (which is same as range) iff a set in its domain is the same as the image of the inverse image of that set.
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) Determine whether the following functions are well-defined:
1a) defined by
1b) defined by
2) Determine whether the function defined by mapping a real number r to the first digit to the right of the decimal point in a decimal expansion of r is well-defined.
3) Apply the Euclidean algorithm to obtain GCD of and express it as a linear combination of 57970 and 10353.
4) For each of the following pairs of integers a and b, determine their greatest common divisor, their least common multiple, and write their greatest common divisor in the form for some integers x and y.
(a) a=20, b=13
(b) a=69, b=372
(c) a=792, b=275
(d) a=11391, b=5673
(e) a=1761, b=1567
(f) a=507885, b=60808
5) Prove that if the integer k divides the integers a and b then k divides for every pair of integers s and t.
6) Prove that if n is composite then there are integers a and b such that a divides ab but n does not divide either a or b.
7) Let a, b and N be fixed integers with a and b non-zero and let be the greatest common divisor of a and b. Suppose and are particular solutions to . Prove for any integer r that integers and are also solutions to (this is in fact the general solution).
8) Determine the value for each integer where denotes the Euler- function.
9) Prove the Well-Ordering Property of integers by induction and prove the minimal element is unique.
10) If p is a prime prove that there do not exist non-zero integers a and b such that (that is, is not a rational number).
11) Let p be a prime and . Find a formula for the largest power of p which divides (it involves the greatest integer function).
12) Prove for any given positive integer N there exist only finitely many integers n with where denotes Euler’s -function.
13) Prove that if d divides n then $latex \phi(d)$ divides where denotes Euler’s -function.
Hope this gives you some math meal to churn for the Pre RMO or PRMO or even the ensuing RMO of Homi Bhabha Science Foundation.
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