------------------------------------------------------------------------------
Mohamed G. Gouda						     CS 311
Summer 2014							     Midterm 3
------------------------------------------------------------------------------

1. (5 points)
Let both sets A and B be the set {0}. Show that 
	PS(A-B) =! (PS(A) - PS(B))
-----------------------------------------------------------------------------

2. (5 points)
Let f:A-->B and g:B-->C be two functions. Use direct inference to prove the
predicate
	((f is surjective) and (g is surjective)) => ((g.f) is surjective)
-----------------------------------------------------------------------------

3. (5 points)
Which of the following functions, if any, is one-to-one and which of them, 
if any is surjective:

	(a) f(x) = (x + 2)
	(b) g(x) = (x^2 + 2)
	(c) h(x) = (x^3 + 2)

Assume that the domains of these functions be the set of all integers.
----------------------------------------------------------------------------

4. (5 points)
Use direct inference to prove that the function f(n) = (n^4 - 2n) is 
Omega(g(n)) where g(n) = (n^4)
----------------------------------------------------------------------------
Solutions:
----------------------------------------------------------------------------

1.
PS(A-B) = {A=B}			PS({})
	= {definition of PS}	{{}}
	=!{definition of a set}	{}
	= {PS(A) = PS(B)}	PS(A) - PS(B)
------------------------------------------------------------------------------

2.
 (z in C)
=>	{g is surjective}
 (Exist y in B, g(y)=z)
=>	{f is surjective}
 (Exist x in A, Exist y in B, f(x)=y and g(y)=z)
=>	{definition of (g.f)}
 (Exist x in A, Exist y in B, (g.f)(x)=z)
=>	{qunatification Exist y in B is not needed}
 (Exist x in A, (g.f)(x)=z)
-------------------------------------------------------------------------------

3.
Function f(x) = x+2 is one-to-one and onto
Function g(x) = (x^2)+2 is neither one-to-one nor onto
Function h(x) = (x^3)+2 is one-to-one but not onto
-------------------------------------------------------------------------------

4.
|f(n)|	=	|n^4 - 2n|
	=	(n^4 - 2n)		for n > 2
	=	(n^4/2 + n^4/2 - 2n)
	>=	(n^4/2)
	=	1/2 * |n^4|
	=	C * |n^4|		for C = 1/2
	=	C * |g(n)|		for K = 2 and C = 1/2
-------------------------------------------------------------------------------