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The problem consists of several logical statements using quantifiers. Let's analyze each statement:

The universal quantifier (∀) means "for all," and the existential quantifier (∃) means "there exists." Here, we need to determine the truth values of the following logical statements under the same universe of discourse as question 3 (not visible in the image).

Let's go through each part:

1. **(∀x)(∀y) (x + y ≤ 10)**:  
   This states that for all values of \( x \) and \( y \), their sum is less than or equal to 10.  
   - The truth of this depends on the set of possible values of \( x \) and \( y \). If the universe of discourse includes all real numbers, this statement is false because, for example, \( x = 10 \) and \( y = 10 \) gives \( x + y = 20 \), which is not less than or equal to 10.

2. **(∃x)(∀y) (x + y = 10)**:  
   This states that there exists a value \( x \) such that for all values of \( y \), the sum of \( x + y \) is always 10.  
   - This statement is false because it implies that a single value of \( x \) should satisfy the equation \( x + y = 10 \) for all values of \( y \), which is not possible. For instance, if \( y = 1 \), \( x + 1 = 10 \) implies \( x = 9 \), but for \( y = 2 \), the same \( x \) would not satisfy the equation.

3. **(∃x)(∀y) (x = y)**:  
   This states that there exists a value \( x \) such that for all values of \( y \), \( x = y \).  
   - This statement is false because there is no single value of \( x \) that can be equal to every \( y \). If \( x = y \), then \( y \) would have to always be the same as \( x \), which is not true for all \( y \).

4. **(∀x)(∃y) (x + y = 10)**:  
   This states that for every \( x \), there exists a \( y \) such that \( x + y = 10 \).  
   - This is true because for any given value of \( x \), you can always find a corresponding \( y \) that satisfies the equation \( x + y = 10 \). For example, if \( x = 4 \), then \( y = 6 \) satisfies the equation.

### Summary of Truth Values:
- (a) False
- (b) False
- (c) False
- (d) True

Would you like further clarification or additional details on any of these statements?

Here are 5 related questions you could explore:
1. What is the difference between universal and existential quantifiers in logic?
2. How does changing the universe of discourse affect the truth value of logical statements?
3. How can logical statements be represented using set theory?
4. What are the conditions under which an existential quantifier can make a statement true?
5. Can a statement with mixed quantifiers (like ∀ and ∃) be simplified logically?

**Tip:** When analyzing logical statements with quantifiers, carefully consider the domain (universe of discourse) and whether the statement applies to all or some elements in that domain.

Tentukan nilai logika kalimat-kalimat berikut berdasarkan penggunaan kuantor universal dan eksistensial pada soal matematika dasar.

Explore how to determine the truth values of logical statements using universal and existential quantifiers in basic mathematical logic. Learn about logical equivalences, quantifier rules, and examples that clarify their use. Suitable for university-level logic or mathematics students.

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