Mohamed G. Gouda						     CS 311
Fall 2013							     Midterm 1

1) Pigeon Hole Principle:
Let m be a non-negative integer and n be a positive integer. 
Prove by contradiction that in any distribution of m pigeons 
into n holes there exist a hole that has at least m/n pigeons.

Sol. 
	not (In any distribution of m piogeons into n holes,
	     Exist a hole that has at least m/n pigeons)
=>	{De Morgan's}
	(Exist a distribution of m pigeons into n holes where 
	All holes have fewer than m/n pigeons each)
=>	{compute number of pigeons}
	number of pigeons < (m/n)*n = m
=>	{number of pigeons = m and m < m is false}
	F

2) Show that the "->" Boolean operator is not associative.

Sol.
We need to exhibit a combination of x, y, and z where the formula 
(x -> (y -> z)) is not equivalent to ((x -> y) -> z).

Consider the combination x=F, y=T, and z=F. In this case, we have
(x -> (y -> z)) = (F -> (T -> F)) = (F -> F) = T
((x -> y) -> z) = ((F -> T) -> F) = (T -> F) = F.

3) Show that the formula f(x,y) = ((x or y) -> (x and y)) is not equivalent
to formula T and not equivalent to formula F.

Sol. 
We exhibit a combination of x and y that makes f(x,y) false and another
combination of x and y that makes f(x,y) true.

f(T,F) = ((T or F) -> (T and F)) = (T -> F) = F
f(T,T) = ((T or T) -> (T and T)) = (T -> T) = T

4) Let Dx be the set of all integers. Use case analysis to prove that
(All x in Dx, x^2 != 8).

Sol.
Case 1: (x >= -2 and x =< 2): in this case, x^2 =< 4 != 8.
Case 2: (x < -2 or x > 2): in this case, x^2 >= 9 != 8.

5) Let k, m, and n be positive integers. Let (k is multiple of m) denote
that k = m*x for some positive integer x. Use direct inference to prove
(All k, m, n, 
((k is multiple of m) and (m is multiple of n)) => (k is multiple of n)).

Sol.
	(k is multiple of m) and
	(m is multiple of n)
=>	{definition of multiple}
	(k = m*x for some positive integer x) and
	(m = n*y for some positive integer y)
=>	{arithmetics}
	(k = n*y*x for some positive integers x and y)
=>	{y*x is a positive integer say z}
	(k = n*z for some positive integer z)
=>	{definition of multiple}
	(k is multiple of n)