Question

1. Prove by induction that, for every natural number n, either 1 = n or 1<n. 2. Prove the validity of the following form of the principle of mathematical in duction, resting your argument on the form enunciated in the text. Let B(n) denote a proposition associated with the integer n. Suppose B(n) is known (or can be shown) to be true when n = no, and suppose the truth of B(n + 1) can be deduced if the truth of B(n) is assumed. Then B(n) is true for every integer n such that no^ n

Answer 1

1. Let P(n) denotes the statement " either 1=n or 1<n ".

Clearly, P(1) is true.

Suppose, P(m) is true. Then, either 1=m or 1<m. Now if 1=m then, 1=m<m+1 and if 1<m then, 1<m<m+1. So P(m+1) is true.

Therefore, P(1) is true and if P(m) is true then P(m+1) is also true. Hence, by induction we proved that P(n) is true for all natural numbers.

2. Suppose
B(no)
is true and the truth of B(n+1) can be deduced if the truth of B(n) is assumed. So, if B(n) is true then B(n+1) is true. Take
n = 1
. We know that
B(no)
is true and so
B(no +1)
is also true. Now take
n= no +1
. Then as
B(no +1)
is true,
Bino +2)
is also true. Again take
n =no + 2
. Then the truth of
Bino +2)
implies the truth of
Bino +3)
. Hence, by continuing this process, we get that B(n) is true for any integer n such that
non
.