Let T be the statement: For all real numbers x and y with x < y, there exists an integer n such that x ≤ n ≤ y.Write the negation of T.

To negate the statement T, we will follow the logical steps necessary to express the opposite of the original statement.

1. ### Identify the Original Statement

The original statement T is: "For all real numbers $x$ and $y$ with $x < y$, there exists an integer $n$ such that $x \leq n \leq y$."

2. ### Break Down the Statement

• The phrase "For all real numbers $x$ and $y$ with $x < y$" is a universal quantifier, which we will negate to an existential quantifier.
• The phrase "there exists an integer $n$ such that $x \leq n \leq y$" is an existential quantifier that we will negate to a universal quantifier.

3. ### Formulate the Negation

The negation can be constructed as follows:

• The phrase "For all $x$ and $y$ with $x < y$" becomes "There exist $x$ and $y$ such that $x < y$."
• The phrase "there exists an integer $n$" becomes "for all integers $n$."

Thus, the negated statement $\neg T$ is reformulated as: "There exist real numbers $x$ and $y$ such that $x < y$ and for all integers $n$, it is not true that $x \leq n \leq y$."

4. ### Final Statement of the Negation

Putting it all together, the negation of statement T is: "There exist real numbers $x$ and $y$ such that $x < y$ and for all integers $n$, $n < x$ or $n > y$."

This means that no integer $n$ can be found between $x$ and $y$, which effectively states that there is no integer within that interval.