Sr no: 000 class: XI sub: Maths chap: 4 Induction lesson:01
***Meaning: The process of reasoning from particular facts or ideas to a general rule or law.
We talk about understanding fundamentals in all our lessons. Half of the understanding comes from understanding clearly the MEANING of the topic itself. So, look up in the dictionary for every topic for its real meaning. Learning starts with that.
***We all know Newton observed an apple falling, discovered some facts and later reasoning turned them into laws.
What is a LAW or a RULE: It means what you say MUST happen, once, twice and ALWAYS. Then only it is a law. We cannot say that because something is true for 100000 times it will be true ALWAYS. Law states this fact that applies without any condition or limitation.
We must find methods to show that it is true always. General methods, methods which do not depend on actual counting but are true for any number in general .
***In chapter on sets, we read about set builder method. Think about it. What did we do? That was our first step to INDUCTION. We found a PATTERN. And then stated this pattern to save space and labor. Then we wrote this pattern in form of a formula.
Remember Newton observed an apple falling and developed laws. What if he had seen an orange? Would the Laws still be true?
We hear people say, “Man is mortal”. Does it mean a “woman is immortal? Here everybody that is born must die. Law!
***If we are thinking of making a law then we must look for a pattern.
Like a natural number. Counting that happens naturally. The difference between this number and the NEXT is only one. We start with 1. We never say 0 cows. Jab hain hi nahin toh ginenge kaise.
Like anything that should come naturally after something.
Or sum of first three natural numbers is = ….. Note NATURAL NUMBERS. NOT integers. NOT real numbers. Because we know that next natural number after 5 is 6. We can define range.
***There is one more similar term: DEDUCTION. This is opposite of Induction.
In Induction we MADE Law from observations.
In Deduction we APPLIED existing law to what we observe. We apply law of mortality to a newly born child. And declare this child will die (someday) because …….?
THEREFORE: for a statement to be a LAW then if we observe something happening to a person at Mumbai must be happening to all persons elsewhere.
NOTE for induction: there must be pattern. We cannot apply induction to a set of numbers like
1,8,22,333,54,.. etc We don’t know what will come next.
PUT simply: If this happened to Ram it must happen to person NEXT to him. We don’t know who is next to Ram. Even if he is Rahim, Albert, Shyam or Sita or anyone else.
Then it becomes LAW.
***We should be able to DEFINE our finding as a FORMULA or relationship. Try with your textbook example.
***We use our knowledge of Induction when we say that “ a statement which is SAID TO BE TRUE for say 1000 (n) occasions will be treated true by a mathematician only when this statement is true for any NATURAL number k ( say 501) within 1000 AND THE NEXT NATURAL number also that is for NEXT number( k+1) ie 502. “
***PROCEDURE SHALL BE:
1. Given a relation understand what it is. Can you put some values, NATURAL NUMBERS, starting with 1 and see if the relation is really correct for 1. We are told that it is correct for some general relationship in terms of n. (say, n is meant to be 1000 or any known to teacher, but not to us).
2. If it works, then choose any NATURAL number k.
You may ask what is special about k and what was not with n. We know nothing about both. Answer is that n is the total number of instances told to us, possibly known to someone. We do not know what happens after n. We have to remain WITHIN n, so k is somewhere inside n. And so is (k+1). WHY because we can only identify natural numbers; we know 6th item will come after 5th, NATURALLY.
3. Since we know nothing about n and k we can later on say n is infinite. And declare the proof as UNIVERSALLY true.
NOW READ YOUR BOOK. IT IS EASY.
Try to do one solved example fully. Then you’ll do all in no time