Truth Tables

[South89]

Here is a question I have from one of my math problems:
Write the argument using symbols and write a conclusion in english which follows from all the premises.
We go to the beach or we do not go to class.
If we do not go to class, then we will not study.
If we do not go to the party, then we will study.
Therefore, ...........
Thanks for the help.

 

[Denis]

What will your teacher say if you turn in something not done by yourself?
Will your teacher sue us?

 

[soroban]

Hello, "South89!

Write the argument using symbols and write a conclusion in English.

. . We go to the beach or we do not go to class.
. . If we do not go to class, then we will not study.
. . If we do not go to the party, then we will study.
. . Therefore . . .

\(\displaystyle \text{Let: }\;\begin{Bmatrix}B: & \text{We go to the beach} \\ C: & \text{We go to class} \\ S: & \text{We study} \\ P: & \text{We go to the party} \end{Bmatrix}\)

\(\displaystyle \text{The premises are: }\;\begin{Bmatrix}B \;\vee \sim C & [1] \\\sim C \to \;\sim S & [2] \\ \sim P \to S & [3] \end{Bmatrix}\)


\(\displaystyle \text{Premise [1] can be written: }\:\sim B \to \sim C\)

\(\displaystyle \text{The contrapositive of [3] is: }\:\sim S \to P\)

\(\displaystyle \text{Then the premises become: }\;\begin{Bmatrix}\sim B \to \;\sim C \\ \sim C \to \;\sim S \\ \sim S \to P \end{Bmatrix}\)


\(\displaystyle \text{By the Law of Syllogism, we have: }\;\sim B \to P\)

. . \(\displaystyle \text}"If we do not go to the beach, then we will go to the party."}\)