(b)
June 3, 1997
1 (a)
(b)
2 (a) 
(b) 
3 (a) No. For example let 
and 
. Then 
.
(b) Take an arbitrary element x in the universe. Then
.
4 (a) Let ODD denote the set of odd integers.
Basis Clause: 
Inductive Clause: If 
, then 
and 
.
Extremal Clause: Nothing is in ODD unless it is obtained from the Basis and Inductive Clauses.
(b) Let ND3 denote the desired set.
Basis Clause: 
Inductive Clause: If 
, then 
.
Extremal Clause: Nothing is in ND3 unless it is obtained from the Basis and Inductive Clauses.
5 (a) Basis Step: Let n = 0. Then LHS = 3*0 = 0, and RHS = 3*0*(0+1)/2 = 0. Hence LHS = RHS.
Inductive Step: 
Since 
by the induction hypothesis,
.
(b) Basis Step: Let n = 1. Then 
and 
.
Since 
by the hypothesis, 
.
Inductive Step: Assume that 
holds and prove
that it holds for k+1.
Take an arbitrary element 
. We try to show that 
.
by the definition of
.
Hence 
.
By the induction hypothesis if 
, then 
.
Also if 
then 
.
Hence 
.
Hence 
.
Hence .