Find the truth value and truth sets of the following statement

(a) forall x \in \mathbb{Z} ,x^ 2 cong0.

(b) forall x \in \mathbb{Z} ,x^ 2 -9=0. "

(c) ^ 4 exists x \in \mathbb{Z} / (x ^ 2) = 3 "

Let's break down each statement step by step, analyzing the truth value and finding the truth set, where applicable:

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### (a) \(\forall x \in \mathbb{Z}, x^2 \cong 0\)
- This statement translates to: "For all \(x\) in the set of integers, \(x^2 \cong 0\)" (modulo some unspecified number).
  
  **Interpretation**:
  - If we assume this congruence means modulo 0, it doesn't make sense because modular arithmetic is undefined for modulo 0. If it's meant to be modulo a different integer (e.g., modulo \(n\)), the statement still isn't true because not all integers squared will be congruent to 0 under modulo \(n\) unless \(x = 0\).

  **Truth value**: **False** (unless \(x = 0\) in a particular modulo context).

  **Truth set**: There is no such truth set because the statement is false for all integers.

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### (b) \(\forall x \in \mathbb{Z}, x^2 - 9 = 0\)
- This statement translates to: "For all \(x\) in the set of integers, \(x^2 - 9 = 0\)", which simplifies to \(x^2 = 9\).
  
  **Interpretation**:
  - The equation \(x^2 = 9\) has two integer solutions: \(x = 3\) and \(x = -3\).
  - However, the statement is asking if **for all** integers, \(x^2 - 9 = 0\). Clearly, this is false because many integers (e.g., \(x = 1\), \(x = 2\)) do not satisfy \(x^2 = 9\).

  **Truth value**: **False**

  **Truth set**: The set of integers that satisfy \(x^2 - 9 = 0\) is \(\{3, -3\}\).

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### (c) \(\exists x \in \mathbb{Z} / (x^2 = 3)\)
- This statement translates to: "There exists an \(x\) in the set of integers such that \(x^2 = 3\)".
  
  **Interpretation**:
  - Solving the equation \(x^2 = 3\), we find that \(x = \pm \sqrt{3}\). However, \(\sqrt{3}\) is not an integer, so there are no integer solutions to this equation.

  **Truth value**: **False**

  **Truth set**: There is no truth set because no integer satisfies \(x^2 = 3\).

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

### Summary of Truth Sets:
- (a) No truth set.
- (b) Truth set: \(\{3, -3\}\)
- (c) No truth set.

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Let me know if you want further clarification or details on any part.

Here are 5 related questions to explore this topic further:
1. How can we define congruence in modular arithmetic?
2. Are there other equations where \(\forall x \in \mathbb{Z}, x^2 - n = 0\) holds?
3. What are the integer solutions to equations of the form \(x^2 = n\) for various values of \(n\)?
4. How do we solve modular arithmetic congruences?
5. Can the equation \(x^2 = 3\) be solved in other number systems like complex numbers?

**Tip**: When dealing with \(\exists\) (there exists) statements, always check for specific solutions, while for \(\forall\) (for all) statements, ensure that the condition holds universally for all elements.

Find the truth value and truth sets of the following statement
(a) forall x ∈ ℤ, x^2 ≡ 0.
(b) forall x ∈ ℤ, x^2 - 9 = 0.
(c) ∃ x ∈ ℤ / (x^2) = 3

Analyze the truth values and truth sets of various mathematical statements involving modular arithmetic and quadratic equations. Learn how to interpret congruence relations and solve for specific integer solutions, suitable for students in Grades 10-12.

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