The problem presented in the image can be broken down into two parts:

Original Proposition:

• Translation: "Because last Saturday and Sunday there was a closing ceremony for PMB 2007, the regular gathering of the Table Tennis Unit (UTM) was canceled, and the TM Open meeting was postponed until today."

(a) Symbolic Notation:

Let:

• $p$: "Last Saturday and Sunday there was a closing ceremony for PMB 2007."
• $q$: "The regular gathering of the Table Tennis Unit (UTM) was canceled."
• $r$: "The TM Open meeting was postponed until today."

The proposition can be written as: $p \rightarrow (q \land r)$ This reads: "If last Saturday and Sunday there was a closing ceremony for PMB 2007 ($p$), then the regular gathering of the Table Tennis Unit (UTM) was canceled ($q$) and the TM Open meeting was postponed until today ($r$)."

(b) Inverse:

The inverse of a proposition $p \rightarrow q$ is $\neg p \rightarrow \neg q$. For this problem:

The inverse of $p \rightarrow (q \land r)$ is: $\neg p \rightarrow \neg (q \land r)$ Which simplifies to: $\neg p \rightarrow (\neg q \lor \neg r)$ This reads: "If there was no closing ceremony for PMB 2007 last Saturday and Sunday ($\neg p$), then either the regular gathering of the Table Tennis Unit (UTM) was not canceled ($\neg q$) or the TM Open meeting was not postponed until today ($\neg r$)."

Would you like a more detailed explanation of the symbolic notation or the inverse? Here are some related questions to think about:

1. What would be the contrapositive of the given proposition?
2. How would you express the converse of the proposition?
3. Can you represent the negation of the entire original proposition?
4. What logical equivalence rules apply to $\neg (q \land r)$?
5. How does the inverse differ from the contrapositive in logic?

Tip:

Always remember that the inverse, converse, and contrapositive of a conditional statement can help explore the relationships between the components of the original proposition.