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 🙂
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) 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
You might want to take a serious shot at each of these. In the first stage of attack, apportion 15 minutes of time for each problem. Do whatever you can, but write down your steps in minute detail. In the last 5 minutes, check why the method or approach does not work. You can even ask — or observe, for example, that if surds are there in an equation, the equation becomes inherently tough. So, as a child we are tempted to think — how to get rid of the surds ?…and so on, thinking in math requires patience and introversion…
So, here are the exercises for your math gym today:
1) Prove that if x, y, z are non-zero real numbers with , then
2) Let a b, c, d be complex numbers with . Prove that
3) Let a, b, c, d be integers. Prove that divides
4) Solve in complex numbers the equation:
5) Solve in real numbers the equation:
6) Find the real solutions to the equation:
7) Solve the equation:
8) Prove that if x, y, z are real numbers such that , then the ratio equals if and only if .
9) Solve in real numbers the equation:
10) Find the real solutions to the system of equations:
PS: if you want hints, do let me know…but you need to let me know your approach/idea first…else it is spoon-feeding…