Answer to Question #173765 in Discrete Mathematics for Ghj

Question

Not having the lecture is sufficient for Dr. Boateng to conduct a quiz 

or students to make presentations. If the students do not make 

presentations then Dr. Boateng will conduct the quiz. Therefore, 

whenever we have the lecture, either the student will not make 

presentations or Dr. Boateng will conduct the quiz.

Answer 1

Denote the following events by

A="Students have the lecture"

B="Students make presentations"

C="Dr. Boateng will conduct the quiz"

Then the conditions of the question may be written in symbols:

$(\bar{A}\to(B\vee C))\wedge (\bar{B}\to C)\to (A\to (\bar{B}\oplus C))$

To determine is this statement a tautology or not, let's solve the equation

$(\bar{A}\to(B\vee C))\wedge (\bar{B}\to C)\to (A\to (\bar{B}\oplus C))=0$. It implies

$(\bar{A}\to(B\vee C))\wedge (\bar{B}\to C)=1$ and $A\to (\bar{B}\oplus C)=0$, it implies

$\bar{A}\to(B\vee C)=1$, $\bar{B}\to C=1$, $A=1$ and $\bar{B}\oplus C=0$, it implies

$A=1, C=\bar{B}$, it implies

$A=1, B=0, C=1$ or $A=1, B=1, C=0$.

If we assume that the events A, B, C are mutually exclusive (i.e. only one of them can take place) then a contradiction is impossible and hence the given logical output is correct.