CS $275-$ Homework $1$

Covers: Chapter $1$ and Chapter $2\mathrm{.}1-2\mathrm{.}4$

Problem $1$:

Using the laws of propositional logic $($Table $1\mathrm{.}5\mathrm{.}1$ in textbook$)$ prove the following:

$A(BC)-=B(AC)$

Problem $2$:

Sherlock Holmes, in "A Study in Scarlet" $($a detective mystery novel by Sir Arthur Conan Doyle$)$ solves the mystery with this quote:

"And now we come to the great question as to the reason why. Robbery has not been the object of the murder, for nothing was taken. Was it politics, then, or was it a woman? That is the question which confronted me$.$ I was inclined from the first to the latter supposition. Political assassins are only too glad to do their work and fly. This murder had, on the contrary, been done most deliberately, and the perpetrator has left his tracks all over the room, showing he had been there all the time."

From these, Sherlock Holmes concluded: $"$It was a woman".

Using the following propositional variables, explain his logic using logical reasoning.

$R$ : It was a robbery.

P: It was politics.

W: it was a woman.

I: An item was taken.

L: The assassin left immediately.

T: Tracks were left.

Hint: The first two lines can be the following:

$RvvPvvW$

$RI$

Problem $3$:

Show a proof by exhaustion for