Let \(P(x,y)\) denote \(x \geq y\text{.}\) Determine whether the following statement are true or false; justify your answer.
\(\forall x, y \in \mathbb{N}, P(x,y)\)
\(\forall x \in \mathbb{N}, \exists y \in \mathbb{N}\) such that \(P(x,y)\)
\(\exists x \in \mathbb{N}\) such that \(\forall y \in \mathbb{N}\text{,}\)\(P(x,y)\)
\(\exists x,y \in \mathbb{N}\) such that \(P(x,y)\)
Checkpoint2.2.2.
Write down the negation of the following statements, moving the negation symbol \(\lnot\) as far to the right as possible. Use symbolic logic notation.
\(\displaystyle \forall x \in A \left [ p(x) \implies q(x) \right ]\)
\(\exists x \in A \) such that \(\left [ \left ( \lnot p(x)\right ) \land q(x) \right ]\)
\(\displaystyle \forall x \in A \left [ p(x) \implies \left ( \exists y \in B \text{ such that } \left [ q(x,y) \land r(x,y) \right] \right ) \right ]\)
Checkpoint2.2.3.
Rewrite the following sentence using symbolic logic notation for the logical connectives and quantifiers (\(\mathcal{F}\) denotes the set of all functions \(\mathbb{R} \to \mathbb{R}\)).
“For any function \(f \in \mathcal{F}\) that is continuous on \([a, b]\text{,}\) there is some \(c \in [a,b]\) such that for all \(x\in [a,b]\text{,}\)\(f(c) \leq f(x)\text{.}\)”
