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Mohamed G. Gouda						     CS 311
Summer 2014							     Midterm 1
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1. (5 points) Let Dx and Dy be the set of all integers. Which of the following 
quantified predicates is T and which of them is F:

a.	(All x, y, (x =< y))

b.	(All x, y, (x =< y) or (y =< x)) = 
	(All x, y, (x =< y)) or (All x, y, (y =< x))

Explain your answers.

2. (7 points) Let x, y, and z be Boolean parameters. Show that there is no 
combination of Boolean values for these parameters that makes the following 
formula f(x, y, z) F.

f(x, y, z) = y or ((x --> (y and not z)) or not y)

(Hint: Show that the two formulas f(x, y, z) and T are equivalent.)

3. (8 points) Let Dx and Dy be the set of all integers. Use direct inference 
to prove the following two predicates:

a.	(All x, y, ((x =< y) and (max(x, y) =< min(x, y)) => x=y)

b.	(All x, y, ((x >= y) and (max(x, y) =< min(x, y)) => x=y)

Solutions

1.

a.	The predicate (All x, y, (x =< y)) is F choose for example x to be 1 and y to be 0

b.	The predicate (All x, y, (x =< y) or (y =< x)) is T
	The predicate (All x, y, (x =< y)) is F from Part a
	The predicate (All x, y, (y =< x)) is F from Part a
	The total predicate is (T = (F or F)) is F 


2.
	f(x, y, z)
= {definition of formula f}
	y or ((x --> (y and not z)) or not y)
= {symmetry and associativity of or}
	(x --> (y and not z)) or (not y or y)
= {idempotence of or}
	(x --> (y and not z)) or T
= {elementary}
	T

3.
a.
	((x =< y) and (max(x, y) =< min(x, y))
=> {definition of max}
	((x =< y) and (y =< min(x, y))
=> {definition of min}
	((x =< y) and (y =< x))
=> {arithmetics}
	(x = y)

	
b.
	((x >= y) and (max(x, y) =< min(x, y))
=> {definition of max}
	((x >= y) and (x =< min(x, y))
=> {definition of min}
	((x >= y) and (x =< y))
=> {arithmetics}
	(x = y)