Expert: Paul Klarreich - 2/12/2010

Question
Dear Sir,
I have seen some statements in my text book. I think these are true statements. But they given in exercise. How to solve these kind of statements. Please do for me.

Using the Principle of Mathematical Induction, prove the following for all n  :
(i)    n  > 1  implies  n - 1 belongs to N.
(ii)    For x belongs to R, with  x  >  0,  if  x  +  n belongs to N  then  x belongs to N .
(iii)     m  +  n,  m n  belongs to N, for  all   m,n belongs to N.

Please answer.
Thanking you,
mahima

Answer
Questioner: mahima
Country: India
Category: Calculus
Private: No
Subject: numbers
Question: Dear Sir,
I have seen some statements in my text book. I think these are true statements. But they given in exercise. How to solve these kind of statements. Please do for me.

Using the Principle of Mathematical Induction, prove the following for all n  :
(i) n  > 1  implies  n - 1 belongs to N.
(ii) For x belongs to R, with  x  >  0,  if  x  +  n belongs to N  then  x belongs to N .
(iii) m  +  n,  m n  belongs to N, for  all   m,n belongs to N.

Please answer.
Thanking you,
mahima
............................................
Look up the principle of mathematical induction.  It will tell you to:

1. WRITE the sentence to be proved for the base case -- the smallest 'n' for which it is to be proved.

-- Now prove that. (usually easy)

2A. WRITE the sentence to be proved for n = k -- the ASSUMPTION case.

2P. WRITE the sentence to be proved for n = k+1 -- the TOPROVE case.

-- Now prove 2P, using 2A.

The most difficult part is, for many, the 'WRITE' part.

However, for a lot of this you need a teacher.