a) No student is present.
b) Albert will not do Mathematics and Albert will not do Physics.
c) The teacher is not teaching or the students are not listening.

d) She is a student and she will not write the exam.
e) For all

a)  If Eric piays golf then Oscar plays tennis.
b)  : if Oscar does not play tennis then Eric plays golf.
c)  : it is not true that Eric plays golf or Oscar plays tennis.
Or: Eric does not play golf and Oscar does not play tennis. (From De Morgan's laws)

We build the truth tables for the two propositions. and compare.
For the proposition  we have

For the proposition  we have

We see that the last columns in the truth tables of  and  are the same.
Hence  and  are
logically equivalent and we write

We build the truth tables for  and  and compare to see if they are exactly the same.
For the truth table of , see solved problem 2 above. Hence we build the truth table for  thus

 We see that the last columns in the truth tables of the propositions  and  are different hence  and  are not logically equivalent.

This proposition has three statements:  and  ; so we shall have 8 possibilities.
We shall first compute  before

i)

 ii) We see that the proposition  is always true.
Hence it is a tautology.

Applying change of base we have