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The problem you provided is asking to determine the truth values of three logical propositions involving the set of numbers \( \{1, 2, 3, 4, 5\} \). Let's analyze each statement.

### The universe of discourse is the set \( \{1, 2, 3, 4, 5\} \).

1. **(a) \( (\exists x) \, x + 1 < 5 \)**  
   This statement is asking if there exists an \( x \) in the set \( \{1, 2, 3, 4, 5\} \) such that \( x + 1 < 5 \).
   
   We will test each element:
   - For \( x = 1 \): \( 1 + 1 = 2 \), and \( 2 < 5 \).
   - For \( x = 2 \): \( 2 + 1 = 3 \), and \( 3 < 5 \).
   - For \( x = 3 \): \( 3 + 1 = 4 \), and \( 4 < 5 \).
   - For \( x = 4 \): \( 4 + 1 = 5 \), and \( 5 \not< 5 \).
   - For \( x = 5 \): \( 5 + 1 = 6 \), and \( 6 \not< 5 \).

   Therefore, at least one \( x \) satisfies the condition (in fact, \( x = 1, 2, 3 \) all work). So the statement is **true**.

2. **(b) \( (\forall x) \, x \leq 5 \)**  
   This statement asserts that for all \( x \) in the set \( \{1, 2, 3, 4, 5\} \), \( x \leq 5 \).
   
   Testing all elements:
   - \( 1 \leq 5 \)
   - \( 2 \leq 5 \)
   - \( 3 \leq 5 \)
   - \( 4 \leq 5 \)
   - \( 5 \leq 5 \)

   All elements satisfy the condition, so the statement is **true**.

3. **(c) \( (\forall x) \, x + 4 < 10 \)**  
   This statement asserts that for all \( x \) in the set \( \{1, 2, 3, 4, 5\} \), \( x + 4 < 10 \).

   We will test each element:
   - For \( x = 1 \): \( 1 + 4 = 5 \), and \( 5 < 10 \).
   - For \( x = 2 \): \( 2 + 4 = 6 \), and \( 6 < 10 \).
   - For \( x = 3 \): \( 3 + 4 = 7 \), and \( 7 < 10 \).
   - For \( x = 4 \): \( 4 + 4 = 8 \), and \( 8 < 10 \).
   - For \( x = 5 \): \( 5 + 4 = 9 \), and \( 9 < 10 \).

   All elements satisfy the condition, so the statement is **true**.

### Summary:
- **(a)** is true.
- **(b)** is true.
- **(c)** is true.

Would you like more details on any specific part? Here are some related questions for practice:

1. How would the truth value of (a) change if the universe of discourse was \( \{3, 4, 5\} \)?
2. What would happen to statement (b) if we changed the upper bound to 4?
3. How would statement (c) change if we modified the expression to \( x + 4 \leq 9 \)?
4. How can we express (b) in negation form?
5. Can we generalize the truth of (b) and (c) for larger sets like \( \{1, 2, \ldots, 10\} \)?

**Tip**: In logic, testing the truth of quantified statements often involves verifying examples for existential quantifiers and checking all cases for universal quantifiers.

Semesta pembicaraan adalah himpunan bilangan 1, 2, 3, 4, 5. Tentukan nilai logika kalimat-kalimat berikut: (a) (∃x) x + 1 < 5, (b) (∀x) x ≤ 5, (c) (∀x) x + 4 < 10.

Explore how to determine the truth values of logical statements using quantifiers (∃x) and (∀x) over the set {1, 2, 3, 4, 5}. This problem covers key logical concepts and inequality operations, suitable for students in Grades 8-10.

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