https://file.aiquickdraw.com/mathbot/file/1729706433743jeslc4ly.jpg

Let's work through the questions in the image.

### 1) Direct Proof:
We are given two statements:
- \( P_1 \): Jika A lulus ujian, maka A mendapat sertifikat (If A passes the exam, then A receives a certificate).
- \( P_2 \): A tidak mendapat sertifikat (A does not receive a certificate).

You are asked to determine the proof method based on these statements. This looks like a scenario for indirect proof, particularly *modus tollens*, where if \( A \) does not receive a certificate, then \( A \) must not have passed the exam. This aligns with \( C \), which likely represents the conclusion, i.e., "A tidak lulus ujian" (A did not pass the exam).

So, in this case:
- \( C \): A tidak lulus ujian (A did not pass the exam).

### 2) Truth Table:
We are asked to determine the truth values for the logical expression:
\[
(P \to \neg Q) \lor (\neg R \land P)
\]
Where \( \to \) denotes implication, \( \lor \) denotes disjunction (OR), \( \land \) denotes conjunction (AND), and \( \neg \) denotes negation (NOT).

Let's define a truth table for this compound expression:
- We will need to calculate truth values for the components \( P \to \neg Q \), \( \neg R \land P \), and their final disjunction.

The table will involve evaluating for all possible values of \( P \), \( Q \), and \( R \) (True and False).

### 3) Matrix Representation:
We are given a relation in the form of a bipartite graph, with elements on the left {2, 3, 4, 1} and elements on the right {36, 24, 28}.

We are asked to convert this to a binary matrix representation and sum each row.

The relation between the elements can be encoded in a binary matrix. For example:
- If an element from the left is connected to an element from the right, we write 1 in the matrix, otherwise 0.

The matrix will look like this:

|     | 36  | 24  | 28  |
|-----|-----|-----|-----|
|  2  |  1  |  1  |  1  |
|  3  |  1  |  1  |  1  |
|  4  |  0  |  1  |  1  |
|  1  |  1  |  0  |  0  |

Now, summing each row:
- Row 1: \( 1 + 1 + 1 = 3 \)
- Row 2: \( 1 + 1 + 1 = 3 \)
- Row 3: \( 0 + 1 + 1 = 2 \)
- Row 4: \( 1 + 0 + 0 = 1 \)

---

### Do you want more detailed calculations or clarifications?

### 5 Related Questions:
1. How can we prove \( C \) (the conclusion) using *modus ponens* instead of *modus tollens*?
2. How do we interpret logical implications (\( P \to Q \)) in a truth table?
3. What other logical identities could simplify the expression in question 2?
4. Can relations be represented as graphs, and how do we transition between different forms (e.g., graph to matrix)?
5. What is the significance of the row sums in a relation matrix?

### Tip:
When working with truth tables, it's useful to break down complex logical expressions step by step, starting from the simplest components.

Tentukan dengan metode Direct Proof:
1) P₁: Jika A lulus ujian, maka A mendapat sertifikat.
   P₂: A tidak mendapat sertifikat.
   C: ...
2) Tentukan dalam Tabel Kebenaran: (P → ¬Q) ∨ (¬R ∧ P)
3) Ubah relasi berikut ke bentuk matriks (unsur 0 dan 1) dan jumlahkan setiap baris binery tersebut.

Solve logical problems using direct proof, truth tables, and binary matrix representations. This includes evaluating compound logical statements and converting relations into matrix form. Ideal for Grades 11-12.

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