The problem involves logical statements and truth values. Let's break it down based on the image provided:

Question 4:

• Given the statement $A : \{(q \Rightarrow s) \land (\neg s \Rightarrow p) \lor (q \lor r) \} \Rightarrow (p \lor r)$

The subquestions are:

(a) Uraikanlah bentuk kalimat A (Break down the form of statement A)

(b) Jika $p = T$ (True), $q = T$, $r = F$ (False), $s = F$, tentukan nilai kebenaran kalimat A. (Determine the truth value of A given specific truth values for $p, q, r, s$)

(c) Tentukan apakah A merupakan tautology, kontingensi, atau bukan keduanya! (Determine whether A is a tautology, a contingency, or neither)

Step-by-step approach:

(a) Uraikanlah bentuk kalimat A:

• The expression can be written in simpler steps: $A : ( (q \Rightarrow s) \land (\neg s \Rightarrow p) \lor (q \lor r) ) \Rightarrow (p \lor r)$

  First, break it into components:

  1. $q \Rightarrow s$ means "if $q$ then $s$".
  2. $\neg s \Rightarrow p$ means "if not $s$, then $p$".
  3. $q \lor r$ means "either $q$ or $r$".
  4. $p \lor r$ means "either $p$ or $r$".

The implication ($\Rightarrow$) will evaluate to false only if the left side is true and the right side is false. So we'll evaluate this after assigning values in part (b).

(b) Truth table calculation:

Let’s substitute the given values:

• $p = T$
• $q = T$
• $r = F$
• $s = F$

We calculate the truth values step by step:

1. $q \Rightarrow s$:

   • $q = T$ and $s = F$, so $q \Rightarrow s = F$.

2. $\neg s \Rightarrow p$:

   • $\neg s = T$ (since $s = F$), and $p = T$, so $\neg s \Rightarrow p = T$.

3. $(q \lor r)$:

   • $q = T$, and $r = F$, so $q \lor r = T$.

Now we combine them:

• $(q \Rightarrow s) \land (\neg s \Rightarrow p)$ becomes $F \land T = F$.
• So $F \lor (q \lor r)$ becomes $F \lor T = T$.

Finally:

• The full expression becomes $T \Rightarrow (p \lor r)$.
• Since $p = T$ and $r = F$, we have $p \lor r = T$.
• So $T \Rightarrow T = T$.

Therefore, the truth value of statement $A$ is True.

(c) Tautology, Contingency, or Neither:

• A tautology is a statement that is always true regardless of the truth values of its components.
• A contingency is a statement that is sometimes true and sometimes false, depending on the values of its components.

We would need to check different truth values of $p, q, r, s$ to determine this. Since we've found that the expression is true for the given values, it could be a contingency or a tautology. To verify whether it’s a tautology, we would need to evaluate the expression for all possible truth values.

Let me know if you want to further evaluate this for all truth values to confirm whether it's a tautology or a contingency!

Questions:

1. How do logical connectives work in different truth tables?
2. Can you identify other statements that would form tautologies?
3. What is the general form of a logical implication?
4. How can truth tables help simplify logical expressions?
5. What are some examples of contingency in daily reasoning?

Tip:

In logic, it's useful to remember that $A \Rightarrow B$ is only false if $A$ is true and $B$ is false, otherwise it's always true.