a. [5 marks] Obtain the sum-of-products. b. [10 marks] Use a K-map to obtain its minimal sum-of-products. Use the table below to supply your answer. c. [10 marks] Use the Boolean identities to show that your result from item a. and b. are equivalent. d. [5 marks] Construct a circuit for your minimized solution using only NOT, 2-input AND, and 2 -input OR gates. 2) [25 marks] For the following Boolean expression provide the following: XYZW+XYZW+XYZW+XYZW+XYZW+XYZW a. [10 marks] Minimize it using the Boolean identities. b. [10 marks] Use K-map to verify the correctness of your solution to item a. Use the table below to supply your answer. c. [5 marks] Construct a circuit for your minimized solution using only NOT, 2-input AND, and 2-input OR gates. 3) [30 marks] You have been asked to construct a digital circuit. This circuit consists of three inputs ( X,Y, and M ) and one output (OUT). When M equals 1 , OUT will equal the Boolean OR of X and Y. If M equals 0 , OUT will equal the Boolean AND of X and Y. a. [5 marks] Construct a truth table to describe the behavior of the circuit. b. [5 marks] Write the sum-of-products for OUT. c. [5 marks] Minimize t using the Boolean identities. d. [10 marks] Use a K-map to obtain the minimal sum-of-products for your OUT result. Use the table below to supply your answer. e. [5 marks] Construct a circuit for your minimized solution. You are to use only NOT, 2-input AND, and 2-input OR gates. 4) [15 marks] Consider the proposition (PQ)(PQ). Show that it is a contradiction by using both: a. [5 marks] the truth table, and b. [10 marks] the logical equivalencies provided by Table 6 below