Demonstrate the validity of the following identities by means of truth table: a. DeMorgan’s theorem for three variables: ( x + y + z)’ = x’y’z’ and (xyz)’ = x’ + y’ + z’ b. The distributive law: x(y + z) = xy + xz c. The associative law and x(yz) = (xy)z

2. Assume you have the following truth tables for functions F₁(x, y, z) and F₂(x, y, z): X y 0 0 0 0 1 1 1 1 oK NOH 0 0 0 1 1 1 0 0 0 1 1 0 1 1 0 F₁ --| 1 1 0 0 0 1 0 0 F2 ононононл 1 1 1 1 a) Express F₁ and F₂ in sum-of-products form b) Simplify each function

Calculate the probability that a harmonic oscillator in the first excited state will be found in the classically forbidden region. (Hint: you will need to numerically integrate using MATLAB). How does your result compare to the result for the ground vibrational state (you don't need to calculate this number because you can find it in the textbook)? How do these probabilities depend on the strength of the bond, te the force constant, kf?

2. (a) Compute and simplify f = (xz+ (y+t)' + (xz)'y')' Encircle final answer! (b) Prove or disprove that in any Boolean algebra, B: (i) 0' = 1 (ii) 1' = 0

2. calculates the trajectory r(t) and stores the coordinates for time steps At as a nested list trajectory that contains [[xe, ye, ze], [x1, y1, z1], [x2, y2, z2], ...]. Start from time t = 0 and use a time step At = 0.01; the last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when z(t) ≤ 0; 3. stores the time for hitting the ground (i.e., the first time t when z(t) ≤ 0) in the variable t_contact and the corresponding positions in the variables x_contact, y_contact, and z_contact. Print t_contact = 1.430 X_contact = 0.755 y contact = -0.380 z_contact = (Output floating point numbers with 3 decimals using format (), e.g., "t_contact = {:.3f}" .format(t_contact).) The partial example output above is for ze = 10. 4. calculates the average x- and y-coordinates 1 y = Yi N where the x, y, are the x(t), y(t) in the trajectory and N is the number of data points that you calculated. Store the result as a list in the variable center = [x_avg, y_avg]…

Please make a k-map to simplify the given boolean equation (and list the final simplified expression)(((((Y AND (~Z)) OR ((~Y) AND Z)) OR (~Z)) AND (~(X OR Y))) OR  (~(((Y AND (~Z)) OR ((~Y) AND Z)) OR (~Z))) AND (X OR Y))