Mohamed G. Gouda						     CS 313K
Fall 2012							     Exercise 1

1)Simplify the following formulas:

  (T and F) or (F and T) or (T and T)
= F or F or T
= T

  (not T or F) and (not F or T) and  not (F or F)
= F and T and T
= F

  not (F or not (T and not(not T or not (F and T))))
= not (F or not (T and not(F or not F)))
= not (F or not (T and not T))
= not (F or not F)
= not T
= F

2)Which of the following formulas are satisfiable?  Explain your answers

  x or (y and not z) or T
is satisfiable because T at the end of the formula makes this formula T

  x or (y and not z) or F
is satisfiable because x can be T which makes this formula T

  x or (not x and y) or (not x and z) or (not y and z)
is satisfiable because x can be T which makes this formula T

  not (x or not F) and (not y or z) and (not z or y)
is not satisfiable because the part not (x or not F) is F which makes this
formula F (regardless of the values of x, y, and z)

  x and (y and not y) and not z
is not satisfiable because the part (y and not y) is F which makes this
formula F (regardless of the values of x, y, and z)

  x and (y or not y) and not z
is satisfiable because this formula is T when x is T and z is F (regardless of
the value of y)
 
  x and not y and z and T
is satisfiable because this formula is T when x is T, y is F, and z is T