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Mohamed G. Gouda						     CS 311
Spring 2015							     Midterm 3
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1. (5 points)
Prove the predicate
	(A sub B) => (A = (A^B))
where A and B are sets and ^ denotes the intersection set operator.
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2. (5 points)
Let f(n) and g(n) be the following two functions. 
	f(n) = 2(n^2)
	g(n) = (n^3) - 2(n^2)

Solve either (a) or (b) below:

(a) Prove that f(n) is O(g(n)).
(b) Prove that g(n) is Omega(f(n)).
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3. (5 points)
Verify that the following program Specification is true
	(T) x := x+y (x+1 > y)
where T is the predicate "true" and
x and y are variables of the program of type nonnegative integers.
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4. (5 points)
Let f:A->A be a function where A is the set of all nonnegative integers and 
f(x) = max(x-1, 0). Show that f(x) is not injective.
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Solutions
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1.
	x in A
<=> {A sub B}
	(x in A) and (x in B)
<=> {definition of ^}
	(x in A^B)
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2.
(a)	|f(n)|	=	|2(n^2)|
		=	2(n^2)			for n > 0
		=<	2(n^2) * (n-2)		for n > 2
		=	2((n^3) - 2(n^2))
		=	C*(g(n))		for C = 2
		=	C*|g(n)|		for K=2 and C=2

(b) 	|g(n)|	=	|(n^3) - 2(n^2)|
		=	((n^3) - 2(n^2))	for n > 2
		>=	(4(n^2) - 2(n^2))	for n > 4
		=	2(n^2)
		=	C*(2(n^2))		for C = 1
		=	C*|f(n)|		for K=4 and C=1
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3.
We need to prove the following predicate:
	(All nonengative integers x and y, (T) => (x+y+1 > y))

A direct inference proof of this predicate is as follows:

	T
=> 	(x >= 0) and (y >= 0)
=>	(x+y) >= y
=>	(x+y+1) > y 
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4.
We can use the "guessing proof style" to prove that f(x) is not injective.

Guess two nonnegative integers x1 and x2 such that f(x1)=f(x2) and x1=!x2.
The guessed x1 and x2 are 0 and 1.
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