Complete Mathematical Induction and Prime Numbers
Original Notes adopted from September 18, 2001 (Week 2)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
Complete Mathematical Induction
If S
⊂
N such that:
a) 1
∈
S
b) (k +1 )
∈
S whenever { 1,2,3, .
.., k }
⊂
S, then S = N.
Example: What is Wrong with this Proof?
Thm: For each n
∈
N, every set of people consists of people the same age.
Proof: True for n = 1.
Use Complete Mathematical Induction
Assume true for n = 1,2,3,.
..k.
Some k.
Consider case n = k +1.
Let S be any set of k +1 people, say S = { x
1
, x
2
,... x
k
, x
k+1
}
Let S
0
= { x
1
, x
2
,... x
k
} k people, so all some age by induction hypothesis.
Let S
1
= { x
2
,x
3
,x
4
, .
.. x
k
, x
k+1
}
k people, all same age.
In particular, all same age as x
2
etc. for elements of S
0
.
∴
Everyone in S same age as x
2
∴
By Mathematical Induction (Complete/Ordinary), every set of people consists of people the
same age.
Series with Positive Terms: