The image contains exercises on propositional logic from "Latihan 4". Here’s a breakdown of the questions:

1. First question: Use laws of propositional logic to show that the following statements are tautologies:

   • (i) $(p \land q) \Rightarrow (p \lor q)$
   • (ii) $[p \land (p \Rightarrow q)] \Rightarrow q$

2. Second question: Use laws of propositional logic to determine the negation and contraposition of the statement:

   • "Dia tidak pergi ke kampus maupun ke perpustakaan bilamana hari ini hujan" (This translates to "He/she does not go to the campus or the library if today is raining").

3. Third question: Show that the statements $(p \Rightarrow q) \Rightarrow r$ and $p \Rightarrow (q \Rightarrow r)$ are not equivalent.

Let's go over how to approach these:

1. First question (tautology proofs):

   • (i) $(p \land q) \Rightarrow (p \lor q)$:
     • By definition, $p \lor q$ (disjunction) covers both the case where either $p$ or $q$ is true. Since $p \land q$ implies both are true, the conclusion $(p \lor q)$ must also hold. This makes the statement a tautology.
   • (ii) $[p \land (p \Rightarrow q)] \Rightarrow q$:
     • The statement assumes both $p$ and the implication $p \Rightarrow q$. By the truth table for implications, when $p$ is true, $q$ must be true for the implication to hold. Hence, the conclusion $q$ is also a tautology.

2. Second question (negation and contraposition):

   • The original statement: "Dia tidak pergi ke kampus maupun ke perpustakaan bilamana hari ini hujan" can be written in symbolic form as: $(\text{hujan} \Rightarrow (\neg \text{ke kampus} \land \neg \text{ke perpustakaan}))$
     • Negation: The negation would be: $\neg (\text{hujan} \Rightarrow (\neg \text{ke kampus} \land \neg \text{ke perpustakaan})) = \text{hujan} \land (\text{ke kampus} \lor \text{ke perpustakaan})$
     • Contraposition: Contrapositive of $(p \Rightarrow q)$ is $(\neg q \Rightarrow \neg p)$: $(\text{ke kampus} \lor \text{ke perpustakaan}) \Rightarrow \neg \text{hujan}$

3. Third question (non-equivalence of statements):

   • For $(p \Rightarrow q) \Rightarrow r$, the truth of $r$ depends on the implication $p \Rightarrow q$.
   • For $p \Rightarrow (q \Rightarrow r)$, the truth of $r$ depends on both $p$ and $q$.
   • These two logical expressions can yield different truth values, and hence are not equivalent.

Would you like further clarification on any part, or should I dive deeper into any specific proof?

Related Questions:

1. What is the difference between a tautology and a contradiction in propositional logic?
2. How do we construct the truth table for complex logical expressions like $(p \Rightarrow q) \Rightarrow r$?
3. Can you explain how the contraposition and the inverse of an implication differ?
4. What is De Morgan's law, and how can it be applied to the negation of conjunctions or disjunctions?
5. How do we determine logical equivalence between two propositions using truth tables?

Tip:

When dealing with logical statements, creating a truth table is a powerful method to visualize and prove statements like tautologies or equivalences.