3.22 Simplify each of the following expressions: (a) xy + x'yz' +yz (c) xy, + z + (x' + y)z, 4.2 A flow rate sensing device used on a liquid transport pipeline functions as follows. The device provides a 5-bit output where all five bits are zero if the flow rate is less than 10 gal- lons per minute. The first bit is I if the flow rate is at least 10 gallons per minute, the first and second bits are 1 if the flow rate is at least 20 gallons per minute; the first, second, and third bits are 1 if the flow rate is at least 30 gallons per minute; and so on. The five bits, represented by the logical variables A, B, C, D, and E, are used as inputs to a device that provides two outputs Y and Z (a) write an equation for the output Y if we want Y to be 1 iff the flow rate is less than 30 gallons per minute. (b) Write an equation for the output Z if we want Z to be 1 iff the flow rate is at least 20 gal- lons per minute but less than 50 gallons per minute. 4.8 A switching circuit has four inputs as shown. A and B represent the first and second bits of a binary number N. C and D represent the first and second bits of a binary number N2.The output is to be 1 only if the product N, X N2 is less than or equal to 2. (a) Find the minterm expansion for F (b) Find the maxterm expansion for F Express your answers in both decimal notation and algebraic form. N, 4.9 Given: Fla, b, c) = abc, + b'. (a) Express F as a minterm expansion. (Use m-notation.) (b) Express Fas a maxterm expansion. (Use M-notation.) 4.16 A bank vault has three locks with a different key for each lock. Each key is owned by a different person. In order to open the door, at least two people must insert their keys into the assigned locks. The signal lines A, B, and C are 1 if there is a key inserted into lock 1, 2, or 3, respectively. Write an equation for the variable Z which is 1 iff the door should open. 5.7 Find the minimum sum-of-products expression for each function. (a) f(a b, c, d) " m (0, 2, 3, 4, 7, 8, 14) (c) fa, b, c, d) = M ( 1, 2, 3, 4, 9, 15)