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The question asks to determine the logical values (truth or falsity) of each statement given that the universe of discourse is the set of real numbers. Let's analyze each statement:

### a. \( (\forall x) \, 2x + 1 = 2x \)
- This statement means that for all \( x \), the equation \( 2x + 1 = 2x \) must hold. Simplifying the equation:
  \[
  2x + 1 = 2x \implies 1 = 0
  \]
  This is clearly false for all real numbers. Hence, the statement is **false**.

### b. \( (\exists x) \, 2x + 1 = 2x \)
- This statement means that there exists some \( x \) such that \( 2x + 1 = 2x \). From the analysis in part (a), we know that the equation reduces to \( 1 = 0 \), which is impossible for any real number. Therefore, this statement is also **false**.

### c. \( (\forall x) \, |x| = x \)
- This statement asserts that for all \( x \), \( |x| = x \). We know that:
  \[
  |x| = x \quad \text{only if} \quad x \geq 0
  \]
  However, for negative \( x \), \( |x| = -x \), so the statement does not hold for all real numbers. Therefore, this statement is **false**.

### d. \( (\exists x) \, |x| = x \)
- This means there exists at least one \( x \) such that \( |x| = x \). This is true for any \( x \geq 0 \). For example, \( x = 1 \) satisfies this condition. Therefore, this statement is **true**.

### e. \( (\exists x) \, |x| = 0 \)
- This means there exists an \( x \) such that \( |x| = 0 \). We know that \( |x| = 0 \) only if \( x = 0 \). Therefore, this statement is **true** since \( x = 0 \) satisfies the condition.

### f. \( (\exists x) \, x^2 = x \)
- This means there exists an \( x \) such that \( x^2 = x \). Simplifying the equation:
  \[
  x^2 = x \implies x(x - 1) = 0
  \]
  So, \( x = 0 \) or \( x = 1 \). Since there are solutions, the statement is **true**.

### Summary of Results:
a. False  
b. False  
c. False  
d. True  
e. True  
f. True

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Would you like more details on any of these solutions or have any other questions?

Here are 5 related questions to expand on the concepts:
1. What is the difference between universal and existential quantifiers in logic?
2. How do absolute values affect equations with variables?
3. Can you explain why \( |x| = x \) is true only for non-negative values of \( x \)?
4. How can we solve equations involving \( x^2 = x \) and similar quadratic forms?
5. What are the implications of contradictory statements like \( 2x + 1 = 2x \) in logical reasoning?

**Tip**: When solving logical statements, always pay close attention to whether the statement is universal (applies to all values) or existential (applies to at least one value).

Semesta pembicaraan adalah himpunan bilangan-bilangan real. Tentukan nilai logikanya kalimat-kalimat di bawah ini.

Explore logical statements involving universal and existential quantifiers over real numbers. Analyze each statement with a detailed explanation, covering absolute value equations and contradictions like 2x + 1 = 2x. Suitable for Grades 10-12.

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