PracticeProofsTest2.pdf - Discrete Math Proof Problems...

Discrete Math
Proof Problems
Inductions
1) For all integers 3 n  ,
( 2)( 3)
3 4 5
2
n n
n
 
     .
2) A sequence
1 2 3
, , ,... a a a
is defined recursively as follows:
1
0
4 1 for all integers 1
2
k k
a a k
a

  

Prove that
7 4 1
3
n
n
a
 

for all integers 0 n  .
3) A function is defined recursively as follows:
4 1
( ) 8 2
3 ( 1) ( 2) 2
n
S n n
S n S n n
 

 


   

Prove that ( ) S n is divisible by 4 for all integers 1 n  .
4) Let f: N → N be a function so that f(a + b) = f(a) * f(b) and f(1) = 2. Prove f(n) = 2 n for all
integer n in N .
5) Define a set X recursively as
B. 2 X 
R. If , then 4 x X x X   
Prove that every element of X is even.
6) Define a recursive function PRD on the set of lists of real numbers by;
1 2 1 2
. ( ) , if
. ( ) ( ) ( ) if ,
PRD L x L x
PRD L PRD L PRD L L L L
 
  
B
R
The set of lists on the real numbers is defined by
1 2 1 2
. for all
. , for all lists ,
L x x
L L L L L
 

B R
R
Set Proofs:
7) For all sets A, B, if A B  , then
A B A  
8) For all sets A, B and C,
 
( ) ( ) A B C A C B C      