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Mohamed G. Gouda						    CS 311
Fall 2014							    Homework 1
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1. (2 points)
Use Formula Equivalence Laws to prove that the following two formulas are 
equivalent
	f(x) and (g(x) and h(x)) 
	(f(x) and g(x)) and (f(x) and h(x))

2. and 3.
Let x and y be two parameters whose domains Dx and Dy, respectively, are the 
set {Amy, Barb, Cindy}.

2. (2 points)
Define a parameterized predicate P(x, y) such the following predicate is T
	(All x, Exist y, P(x, y)) = (Exist y, All x, P(x, y)) 
Explain your answer.

3. (2 points)
Define a parameterized predicate P(x, y) such the following predicate is T
	not (Exist y, P(Barb, y))
Explain your answer.

4. and 5.
Consider the following three definitions:

A non-negative integer x is called 0-third iff x = 3m for some non-negative
integer m.

A non-negative integer x is called 1-third iff x = 3m+1 for some non-negative
integer m.

A non-negative integer x is called 2-third iff x = 3m+2 for some non-negative
integer m.

4. (2 points)
Let Dx be the set of all non-negative integers, use direct inference to 
verify the following predicate
	(All x, (x is 2-third) ==> (x^2 is 1-third))

5. (2 points)
Let Dx and Dy be the set of all non-negative integers, use direct inference to 
verify the following predicate
	(All x, y, ((x is 1-third) and (y is 2-third)) ==> (xy is 2-third))
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Solutions:
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1.
	f(x) and (g(x) and h(x)) 
==> {idempotence}
	(f(x) and f(x)) and (g(x) and h(x))
==> {associativity}
	f(x) and (f(x) and g(x)) and h(x)
==> {symmetry}
	f(x) and (g(x) and f(x)) and h(x)
==> {associativity}
	(f(x) and g(x)) and (f(x) and h(x))

2.
Define P(x, y) as follows:
	P(Amy, Amy)	= T
	P(Barb, Amy)	= T
	P(Cindy, Amy)	= T
	P(x, y)		= F 	for all other x and y
In this case, 	(All x, Exist y, P(x, y)) = T and
		(Exist y, All x, P(x, y)) = T 

3.
Define P(x, y) as follows:
	P(Barb, Amy)	= F
	P(Barb, Barb)	= F
	P(Barb, Cindy)	= F
	P(x, y)		= T	for all other x and y
In this case,	not(Exist y, P(Barb, y)) =
		(All y, not(P(Barb, y))) = T

4.
	(x is 2-third) 
==> {definition of 2-third}
	x = (3m+2)	for some non-negative integer m
==> {compute x^2}
	x^2 = (3m+2)^2 	for some non-negative integer m
==> {analysis}
	x^2 = (9m^2 + 12m + 4)
			for some non-negative integer m
==> {analysis}
	x^2 = 3(3m^2 + 4m + 1) + 1
			for some non-negative integer m
==> {(3m^2 + 4m + 1) is non-negative integer}
	x^2 = 3n + 1	for some non-negative integer n
==> {definition of 1-third}
	x^2 is 1-third

5.
	((x is 1-third) and (y is 2-third))
==> {definitions of 1-third and 2-third}
	x = 3m+1 and y = 3n+2	for some non-negative inetgers m and n
==> {compute xy}
	xy = (9mn + 6m + 3n + 2)
				for some non-negative inetgers m and n
==> {analysis}
	xy = (3(3mn + 2m + n) + 2)
				for some non-negative inetgers m and n
==> {(3mn + 2m + n) is a non-negative integer}
	xy = (3r + 2)
				for some non-negative inetger r
==> {definition of 2-third}
	xy is 2-third
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