15.1, problems
1a
Find the value of each of the following Boolean expressions if the values of the Boolean variables w, x, y, and z are 1, 1, 0, and 0, respectively. xy + x y x y xy (xy) ̅ x ̅ y ̅ x ̅y ̅ (xy) ̅+x ̅y ̅
1 0 0 1 0 1 0 1
Solution: 1

2a
Let w, x, and y be Boolean variables where the value of x is 1. For each of the following Boolean expressions, determine, if possible, the value of the expression. If you cannot determine the value of the expression, then find the number of assignments of values for w and y that will result in the value 1 for the expression.
a) x + xy + w x = 1 then x+xy+w = 1+1y+w = 1+(1y+w)
1+z = 1 for every Boolean value z then: if we take z = 1y+w
1+(1y+w) = 1
Solution: Yes is possible, x+xy+w = 1

4a
a) Find the fundamental conjunction made up from the variables w, x, y, z, or their complements, where the value of the conjunction is 1 precisely when
i) w _ x _ 0, y _ z _ 1. ii) w _ 0, x _ 1, y _ 1, z _ 0. iii) w _ 0, x _ y _ z _ 1. iv) w _ x _ y _ z _ 0.

9
Let Bn →B. If the d.n.f. of f has m fundamental conjunctions and its c.n.f. has k fundamental disjunctions, how are m, n, and k related?

12
Find the values of the Boolean variables w, x, y, z that satisfy the following system of simultaneous (Boolean) equations. x + xy _ 0 xy _ xz xy + x z + zw _ zw x+x ̅y=0→(x+x ̅ )(x+y)=0 Distributive Law of + over 
1(x+y)=0 Inverse Law of + x+y =0 Identity law of  x =0 and y = 0 Property of +
Since x ̅y=x ̅z→0 ̅0=0 ̅z→10=1z→0=1z→0=z
Then z =0
SInce: x = y = z = 0 and x ̅y+x ̅z ̅+zw=z ̅w
0 ̅0+0 ̅0 ̅+0w=0 ̅w→10+11+0=1w
0+1+0=w
[0+1]+0=w
1+0=w
1=w