Question

13 multiple choice 1 point consider a school with two periods during which classes can be scheduled. each class has two variables. for example, class a has variables $x_{a1}$ and $x_{a2}$. setting $x_{a2}=1$ represents scheduling class a during period 2. select the boolean expression that is true if and only if class a is scheduled during exactly one of the two periods. $(x_{a1}+x_{a2})(overline{x_{a1}}overline{x_{a2}})$ $(x_{a1}x_{a2})(overline{x_{a1}+x_{a2}})$ $(x_{a1}x_{a2})(overline{x_{a1}}+overline{x_{a2}})$ $(x_{a1}+x_{a2})(overline{x_{a1}}overline{x_{a2}})$ 14 multiple choice 1 point select the boolean expression that corresponds to the output of the boolean circuit below: $x$ $y$ output $z$ $overline{x}+y+yz$ $overline{x}overline{y}(y + z)$ $x + y+overline{y}z$ $overline{x + y}+overline{y}z$

Explanation:

Step1: Analyze question 13

The condition for class $A$ to be scheduled in exactly one of the two periods means either it is scheduled in period 1 and not in period 2 or it is scheduled in period 2 and not in period 1. In Boolean - logic, this is equivalent to $(x_{A1}\overline{x_{A2}}+\overline{x_{A1}}x_{A2})=(x_{A1} + x_{A2})(\overline{x_{A1}}\overline{x_{A2}})$ using the distributive law of Boolean algebra.

Step2: Analyze question 14

1. First, the output of the OR - gate with inputs $x$ and $y$ is $x + y$.
2. Then, the output of the NOT - gate on the output of the OR - gate is $\overline{x + y}$.
3. The output of the AND - gate with inputs $y$ and $z$ is $yz$.
4. The final output of the OR - gate with inputs $\overline{x + y}$ and $yz$ is $\overline{x + y}+yz$.

Answer:

1. $(x_{A1} + x_{A2})(\overline{x_{A1}}\overline{x_{A2}})$
2. $\overline{x + y}+yz$