2. Consider the following logical propositions
a : ∀x ∈ Z, ∃y ∈ N0 : x2 = y
The ¬a and logical value of a are:
(a) a = T and ¬a = ∀x ∈ Z, ∃y ∈ N: x2 6= y
(b) a = T and ¬a = ∃x ∈ Z, ∀y ∈ N: x2 6= y X
(c) a = F and ¬a = ∃x ∈ Z, ∀y ∈ N: x2 6= y

Let's analyze the given logical proposition \( a \) and its negation \( \neg a \) to determine the correct answers.

### Proposition \( a \):
\[ a : \forall x \in \mathbb{Z}, \exists y \in \mathbb{N}_0 \text{ such that } x^2 = y \]

This statement means that for every integer \( x \), there exists a non-negative integer \( y \) such that \( x^2 = y \).

- For any integer \( x \), \( x^2 \) is always a non-negative integer. Therefore, for each \( x \in \mathbb{Z} \), we can always find a \( y \in \mathbb{N}_0 \) (specifically \( y = x^2 \)) such that \( x^2 = y \).
- This makes the proposition \( a \) true (\( a = T \)).

### Negation \( \neg a \):
To find the negation \( \neg a \), we apply logical negation:

\[ \neg a : \exists x \in \mathbb{Z} \text{ such that } \forall y \in \mathbb{N}_0, x^2 \neq y \]

This means there exists some integer \( x \) such that for all non-negative integers \( y \), \( x^2 \neq y \). However, this statement is false because \( x^2 \) will always equal some \( y \in \mathbb{N}_0 \).

### Comparing with the Options:
Now let's compare the analysis with the given options:

- **(a)** \( a = T \) and \( \neg a = \forall x \in \mathbb{Z}, \exists y \in \mathbb{N}: x^2 \neq y \)  
  This option is incorrect because the negation is incorrectly formulated.

- **(b)** \( a = T \) and \( \neg a = \exists x \in \mathbb{Z}, \forall y \in \mathbb{N}_0: x^2 \neq y \)  
  This option is correct in stating \( a = T \) and the correct formulation of the negation \( \neg a \).

- **(c)** \( a = F \) and \( \neg a = \exists x \in \mathbb{Z}, \forall y \in \mathbb{N}_0: x^2 \neq y \)  
  This option is incorrect because \( a \) is not false.

### Correct Answer:
The correct answer is **(b)**.

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the logical negation of the statement \( \exists x \in \mathbb{Z}, \forall y \in \mathbb{N}_0: x^2 = y \)?
2. How does the truth value of a proposition affect the truth value of its negation?
3. Can you provide an example of a proposition involving \( \forall \) and \( \exists \) that is false?
4. What is the difference between \( \mathbb{N} \) and \( \mathbb{N}_0 \) in terms of set theory?
5. How would the proposition change if we were working with \( \mathbb{R} \) instead of \( \mathbb{Z} \)?

**Tip:** When working with logical propositions, always be careful to apply negations correctly, especially when dealing with quantifiers like \( \forall \) (for all) and \( \exists \) (there exists).

Explore the logical proposition ∀x ∈ Z, ∃y ∈ N0 such that x^2 = y and its negation. Learn the correct logical formulations and their implications. Suitable for advanced mathematics enthusiasts and students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation