Q1: Show an algebraic expression for the following circuit. Reduce it to a minimum sum of products expression (2 terms, 4 literals.) c a y b' w g c' d
z x
w = (a b')' = a' + b y = c w = c (a b')' x = c' + d z = w' + x' = a b' + c d' g = (y + z)' = (c w + w' + x') = (c + w' + c d')' = (c + w')' = (c + a b')' = c' (a' + b) = a' c' + b c' ii. Convert the following to a sum of minterms. Remove any duplicates from the list. f = (a + b' + c) (a + c + d) (b + c') (b' + d') = (a + c + b' d) (b d' + b' c') = a b d' + a b' c' + b c d' + b' c' d = a b c' d' + a b c d' + a b' c' d' + a b' c' d + a' b c d' + a b c d' = a' b' c' d + a' b c d' + a b' c' d' + a b' c' d + a b c' d' + a b c d' Q2: Reduce the expression below (using the algebraic properties) to a sum of products expression with 2 terms and 3 literals. Show each step. x' y' z' + x' y' z + x' y z' + x' y z + x y z' = = = = =
x' y' (z' + z) + x' y (z' + z) + x y z' x' y' + x' y + x y z' x' (y' + y) + x y z' x' + x y z' [a + a’ b = a + b] x' + y z'
Consider the following function g = a b + b c' + a c' All inputs are available both uncomplemented and complemented. a) Show a block diagram for a two-level AND-OR implementation.
+
a b b c'
g
a c'
b) Show a block diagram for an implementation that uses only two-input AND and OR gates. g = b (a + c') + a c' or g = c' (a + b) + a b a c'
b
g a c'
Q3: For the following truth table x 0 0 0 0 1 1 1 1
y 0 0 1 1 0 0 1 1
z 0 1 0 1 0 1 0 1
f 1 0 1 1 0 0 0 1
a) Write a sum of minterms function in numeric form (for example, ∑ m (0, ...) f (x, y, z) = ∑ m (0, 2, 3, 7) b) Write a sum of minterms function in algebraic form (for example, x' y + ...) f = x' y' z' + x' y z' + x' y z + x y z c) Find a minimum sum of products expression (2 terms, 4 literals) f = x' z' (y' + y) + y z (x' + x) = x' z' + y z