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Practice Exercise 1. 1. Write a boolean expression describing the output of this circuit. Using De Morgan's law derive a simplified expression and draw an equivalent circuit representing this expression. 2. Prove that (A + B)(A + C) = A + (BC) by the truth table. 3. Show how the AND function can be constructed from two NAND gates. 4. (Multiple Choice) Select the Boolean expression that is not equivalent to x·x+x·x0 (a) x · (x + x0) (b) (x + x0) · x (c) x0 (d) x 2. Select the expression which is equivalent to x · y + x · y · z (a) x · y (b) x · z (c) y · z (d) x · y · z 3. Select the expression which is equivalent to (x + y) · (x + y0) (a) y (b) y0 (c) x (d) x0 4. Select the expression that is not equivalent to x · (x0 + y) + y (a) x · x0 + y · (1 + x) (b) 0 + x · y + y (c) x · y (d) y 5. Convert the following expressions into sum-of-products and product-of-sums forms: (a) (AB + C)(B + CD) (b) X + X(X* +Y)(Y + Z*) 6. Simplify the following Boolean function by K-map f(x,y,z) = m(0,2,4) + d(1,3) 7. Identify the most simple SOP expression which generates the Karnaugh map shown: AB + C 8. Use DeMorgan's theorem to compute the complement of the following Boolean expression: A (B + CD) 9.Use Karnaugh maps (K-maps) to simplify the following functions in sum of products form. How many literals appear in your minimized solutions? (X,Y,Z) = P M(0,1,6,7) (W,X,Y,Z) = P M(1,3,7,9,11,15) (V,W,X,Y,Z) = P M(0,4,18,19,22,23,25,29) (A,B,C,D) = S m(0,2,4,6) (A,B,C,D) = S m(0,1,4,5,12,13) (A,B,C,D,E) = S m(0,4,18,19,22,23,25,29) 10. (Combinational Logic Design) Consider a four-input function that outputs a 1 whenever an odd number of its inputs are 1. Fill in the truth table for the function. Fill in the K-map to \xde nd the minimum sum of products expression for the function. What is it? Can the function be minimized using the K-map method? Can you think of a more economical way to implement this function if XOR gates are allowed? (Warning: It will be very tedious to try to simplify this function using Boolean algebra, so think about the question first!) 11. Write the truth table of the following function 12. Convert 123410 in base 12 13. Use and , or, not gates design the following function f(x,y,z) = x (y+z).