2. Consider the following logical propositions

b : ∀x ∈ N, ∃y ∈ Z : x+y=5

The ¬b and logical value of a are:


Let's analyze the logical proposition \( b \) and its negation \( \neg b \) to determine their logical values.

### Proposition \( b \):
\[ b : \forall x \in \mathbb{N}, \exists y \in \mathbb{Z} \text{ such that } x + y = 5 \]

This statement means that for every natural number \( x \), there exists an integer \( y \) such that \( x + y = 5 \).

### Analysis of \( b \):
- For any \( x \in \mathbb{N} \), we need to find a \( y \in \mathbb{Z} \) such that the equation \( x + y = 5 \) is true.
- Solving for \( y \), we get \( y = 5 - x \).
- Since \( y = 5 - x \) will always yield an integer (because \( \mathbb{Z} \) includes all integers, both positive and negative), such a \( y \) will always exist for any \( x \in \mathbb{N} \).

Therefore, the proposition \( b \) is **true** (\( b = T \)).

### Negation \( \neg b \):
The negation of \( b \) is:

\[ \neg b : \exists x \in \mathbb{N} \text{ such that } \forall y \in \mathbb{Z}, x + y \neq 5 \]

This means that there exists at least one natural number \( x \) such that for all integers \( y \), the equation \( x + y = 5 \) does not hold.

However, since for every \( x \in \mathbb{N} \), there is a \( y \in \mathbb{Z} \) such that \( x + y = 5 \), the negation \( \neg b \) is **false** (\( \neg b = F \)).

### Conclusion:
- The logical value of \( b \) is **true** (\( b = T \)).
- The negation \( \neg b \) is **false** (\( \neg b = F \)).

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the proposition \( b \) change if we replaced \( \mathbb{Z} \) with \( \mathbb{N} \) for \( y \)?
2. What is the impact of changing the domain of \( x \) from \( \mathbb{N} \) to \( \mathbb{Z} \) in the proposition \( b \)?
3. How would you negate a proposition where both \( x \) and \( y \) are quantified by \( \forall \) instead of \( \forall \) and \( \exists \)?
4. What does the negation of a proposition involving universal quantifiers tell us about the original statement?
5. Can you formulate a similar proposition where \( \neg b \) would be true?

**Tip:** When working with logical quantifiers, always consider the domains of the variables carefully, as they affect the truth values and the negation of the proposition.

Explore the logical proposition b: ∀x ∈ N, ∃y ∈ Z such that x + y = 5 and its negation ¬b in depth. Understand the implications of quantifiers in logic and number theory. Suitable for undergraduate-level study.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation