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Let's break down and solve the logical statements based on the universe of discourse \(\{1, 2, 3, 4, 5\}\):

### Given Statements:
1. **The universe of discourse** is \(\{1, 2, 3, 4, 5\}\).
2. Determine the truth values of the following:
   - a. \((\exists x). x + 1 < 5\)
   - b. \((\forall x). x \leq 5\)
   - c. \((\forall x). x + 4 < 10\)

### Solutions:

#### a. \((\exists x). x + 1 < 5\)
- This statement means: "There exists an \(x\) in the set \(\{1, 2, 3, 4, 5\}\) such that \(x + 1 < 5\)."
- Let's check each value of \(x\):
  - \(x = 1 \rightarrow 1 + 1 = 2 < 5\) (True)
  - Since \(x = 1\) satisfies the condition, the statement is **true**.

#### b. \((\forall x). x \leq 5\)
- This statement means: "For all \(x\) in the set \(\{1, 2, 3, 4, 5\}\), \(x \leq 5\)."
- Checking all values:
  - \(x = 1, 2, 3, 4, 5\), all of these satisfy \(x \leq 5\).
- Therefore, the statement is **true**.

#### c. \((\forall x). x + 4 < 10\)
- This statement means: "For all \(x\) in the set \(\{1, 2, 3, 4, 5\}\), \(x + 4 < 10\)."
- Checking each value of \(x\):
  - \(x = 1 \rightarrow 1 + 4 = 5 < 10\) (True)
  - \(x = 2 \rightarrow 2 + 4 = 6 < 10\) (True)
  - \(x = 3 \rightarrow 3 + 4 = 7 < 10\) (True)
  - \(x = 4 \rightarrow 4 + 4 = 8 < 10\) (True)
  - \(x = 5 \rightarrow 5 + 4 = 9 < 10\) (True)
- Since all values satisfy the condition, the statement is **true**.

### Summary of Truth Values:
- a. True
- b. True
- c. True

Would you like any further details or explanations on any part of this solution?

Here are 5 related questions that expand upon this topic:
1. What is the difference between \(\exists x\) and \(\forall x\) in logic?
2. How would the truth values change if the universe was \(\{1, 2, 3, \ldots, 10\}\)?
3. Can a statement with \((\exists x)\) be false if at least one value of \(x\) satisfies the condition?
4. How can you negate statements with \((\forall x)\) and \((\exists x)\)?
5. What other mathematical operations can be used in logical statements besides addition?

**Tip:** In logic, when you see \((\exists x)\), you only need to find one value that satisfies the condition to make the statement true.

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.

Determine the truth values of logical statements using existential and universal quantifiers in the set {1, 2, 3, 4, 5}. This exercise helps students understand propositional logic and inequalities, suitable for Grades 10-12.

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