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Mohamed G. Gouda						    CS 311
Spring 2015							    Homework 1
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1. (2.5 points)

a. Use equivalence laws to prove that the following two formulas are equivalent:
   	f(x,y)	=	x and (x=y)
	g(x,y)	=	x and y

b. Use equivalence laws to prove that the following two formulas are equivalent:
   	f(x,y)	=	x or (x=y)
	g(x,y)	=	x or (not y)

2. (2.5 points)
Let Dx be the set {1, 2} and Dy be the set {a, b}

a. Define the parameterized predicate P(x,y) such that the following quantified 
predicate is T:
	(All x, Exist y, P(x, y))

b. Define the parameterized predicate P(x,y) such that the following quantified 
predicate is T:
	(All x, All y, not(P(x, y)))

c. Define the parameterized predicate P(x,y) such that the following quantified 
predicate is F:
	(All x, Exist y, P(x, y))

3. (2.5 points)
Let Dx be the set {1, 2, ...}. Use case analysis to prove that the Boolean 
value of the following quantified predicate is F:
	(Exist x,  x^3 = 120)

4. (2.5 points)
Let Dx and Dy be the set of all integers. Write a direct inference proof to
prove the following quantified predicate:
	(All x, All y, T => (max(x,y) >= x))

You can use the following hints in your proof:

{H1}:	{(All x, All y, (max(x,y)=x) --> (max(x,y)>=x))}
{H2}:	{(All x, All y, (y>=x) --> (max(x,y)>=x))}
{H3}:	{(All x, All y, (x>=y) --> (max(x,y)=x))}
{H4}:	{(All x, All y, (x>=y) or (y>=x))}
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Solutions:
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1.a.
  	f(x,y)	
= {definition of f(x,y)}	
	x and (x=y)
= {definition of =}
	x and ((x and y) or (not x and not y))
= {distribution of and over or and associativity of and}
	(x and x and y) or (x and not x and not y)
= {idempotence of and, idempotence of and, and elementary}
	(x and y) or (F)
= {elementary}
	(x and y)
= {defintion of g(x,y)}
	g(x,y)

1.b.
  	f(x,y)	
= {definition of f(x,y)}	
	x or (x=y)
= {definition of =}
	x or ((x and y) or (not x and not y))
= {associativity of or}
	x or (x and y) or (not x and not y)
= {absorption}
	x or (not x and not y)
= {distribution of or over and}
	(x or not x) and (x or not y)
= {idempotence of or}
	T and (x or not y)
= {elementary}
	x or not y
= {definition of g(x,y)}
	g(x,y)
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2.a.
For (All x, Exist y, P(x,y)) to be T, define P(x,y) as follows:
	P(1,a)= T  P(1,b)= don't-care  P(2,a)= don't-care  P(2,b)= T

2.b.
For (All x, All y, not P(x,y)) to be T, define P(x,y) as follows:
	P(1,a)= F  P(1,b)= F  P(2,a)= F  P(2,b)= F

2.c.
For (All x, Exist y, P(x,y)) to be F, define P(x,y) such that
	(Exist x, All y, not P(x,y)) is T 
Thus define P(x,y) as follows:
	P(1,a)= F  P(1,b)= F  P(2,a)= don't-care  P(2,b)= don't-care
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3. 
To prove (Exist x, x^3 = 120) is F, one needs to prove (All x, x^2 != 120) is T
which can be proven by case analysis as follows:

Step 1: Partition Dx into two subsets d1 and d2 where
	d1 = (x =< 4)
	d2 = (x >= 5)

Step 2:
	(All x, x =< 4, x^3 =< 64 != 120) is T
	(All x, x >= 5, x^3 >= 125 != 120) is T

Thus	(All x, x in Dx, x^3 != 120) is T
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4.
A direct inference proof of the quantified predicate 

	(All x, All y, T => (max(x,y) >= x))

proceeds as follows

	T
=> {H4: (All x, All y, (x>=y) or (y>=x))}
	(x>=y) or (y>=x)
=> {H3: (All x, All y, (x>=y) --> (max(x,y)=x))}
	(max(x,y)=x)) or (y>=x)
=> {H2: (All x, All y, (y>=x) --> (max(x,y)>=x))}
	(max(x,y)=x) or (max(x,y)>=x)
=> {H1: (All x, All y, (max(x,y)=x) --> (max(x,y)>=x))}
	(max(x,y)>=x) or (max(x,y)>=x)
=> {idempotence of or}
	max(x,y)>=x
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