Let's go through each question one by one and tackle them:

Question 1

Use logical operators (conjunction, disjunction, implication, biconditional, negation, xor, nand, nor): 1a. Jika kamu tidak lapar dan kenyang maka perutmu akan merasa lapar
This translates to:

• Let $p$ be "kamu tidak lapar" (you are not hungry).
• Let $q$ be "kenyang" (full).
• Let $r$ be "perut merasa lapar" (stomach feels hungry).

The statement can be written as:
$(p \land q) \rightarrow r$

Question 2

Construct a truth table for the proposition:
$(p \rightarrow \neg q) \oplus (\neg p \leftrightarrow r)$
Where:

• $\rightarrow$ denotes implication,
• $\neg$ denotes negation,
• $\oplus$ denotes exclusive or (xor),
• $\leftrightarrow$ denotes biconditional.

To solve this, we need to list out all possible truth values for $p$, $q$, and $r$, and then evaluate each component of the expression to complete the truth table.

Question 3

Determine whether the statement below is a tautology, contradiction, or contingent: $p \lor (q \land r) \Rightarrow (p \land \neg q)$

1. A tautology is a statement that is always true.
2. A contradiction is a statement that is always false.
3. A contingent statement is one that is true for some values and false for others.

Question 4

Prove the logical equivalence of the following expressions: $(\neg p \lor q) \lor r \equiv (\neg (p \land q)) \lor r$ This requires using laws of logical equivalences, such as De Morgan's laws, distribution, and associative laws, to show both sides of the equivalence are equal.

Would you like me to go through each question in detail, including truth tables, proofs, or explanations?

Related Questions to Explore Further

1. How does one determine if a complex logical expression is a tautology or a contradiction?
2. What are the different methods to construct a truth table for multiple propositions?
3. How does De Morgan’s law apply in simplifying logical expressions?
4. What are the key steps to proving logical equivalences?
5. How can exclusive or (XOR) be expressed in terms of basic logical operators?

Tip:

When working with truth tables, listing all possible truth values systematically for each proposition ensures accuracy in evaluating complex expressions.