# How is the following an equivalent proposition?

I am taking a discrete mathematics course and one of the questions asks to develop an English translation of a proposition.

The proposition has the following structure [Question 9 part H is where I am having issues]:

I posted the question on the course's Piazza page and a student answered this:

¬p∧(p∨¬q) {My original proposition}
≡
(¬p∧p)∨(¬p∧¬q)


Then saying:

(¬p∧p)∨(¬p∧¬q)
=
(p∨¬q) {The proposition that the student claims is == my original proposition.}


Obviously (p∨¬q) would be simpler to put in words than ¬p∧(p∨¬q). Finding an English statement for ¬p∧(p∨¬q) would be difficult as compared to (p∨¬q).

Is this true? And if so, why?

Swimming at the North Shire is not allowed [$\lnot p$] and (either Swimming at the North Shire is allowed or Sharks have not been spotted near the shore [$p \lor \lnot q$])

is quite "impossible" as an English (or other natural language) sentence.

Thus, if tautological transformations (truth equivalence) are allowed, it is correct that (by Distributivity):

$\lnot p \land (p \lor \lnot q)$

is equivalent to :

$(\lnot p \land p) \lor (\lnot p \land \lnot q)$.

Now (see : Table of Logical Equivalences) :

$\lnot p \land p \equiv FALSE$

and :

$FALSE \lor \alpha \equiv \alpha$.

Thus, the complete formula is equivalent to :

$\lnot p \land \lnot q$.