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HW 2 - 1 a Label the inputs so that the output is a.c(a AND...

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1. a) Label the inputs so that the output is a.c (a AND c) b) Using only AND, OR, NOT gates, draw the combinational logic network for h which precisely corresponds to the following expression. Use fanout as appropriate, but do not simplify . h(w,x,y,z) = (w + x' + y) + 0 y' + (x' z + w')' Assume that complemented inputs are available; i.e., assume that for each variable w, x, y, z, both the variable and its complement are available as inputs: w, w', x, x', y, y', z, z'. 2. a) Consider the following 3-input XOR gate z a b c Write the formula for z in terms of AND, OR, and NOT. b) If any ONE of a, b or c is inverted, what is the new z ? Show that this new z remains the same irrespective of which input you choose to invert. This Boolean expression corresponds to the output of a different type of gate; what is this gate called? c) What happens to z if they are all inverted at the same time? 3. a) A literal is a variable in its true or complemented form. Simplify the following functions such that there are minimum total literals. Write the Boolean identity you are using at each step. Use no more than 8 steps per item. i) ( , , ) ( )( )( ) f x y z x y x y z x y z ± ± ± ± ± ii) ( , , , ) f w x y z x xyz xyz xy wx wx ± ± ± ± ± iii) f(A,B,C,D) = (AB + A’B’)(C’D’ + CD) + (AC)’
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iv) f(X, Y, Z) = X+Y(Z+(X+Z) ) b) (i) Use less than 6 steps to prove: a' + wam' + wm + v'wz = a' + w (ii) State and prove the dual of the theorem in part (i). Align the dual proof
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