Quantifiers: How to Prove/Disprove

Fun class activities: For each of the following quantified propositions, determine whether the statement is true or false. If it is true, provide a proof. If it is false, give a counterexample.

1. Consider the set of natural numbers \( \mathbb{N} \).
   Determine the validity of the following statement: $\forall x \in \mathbb{N}, x + x \geq x.$
2. Consider the set of integers \( \mathbb{Z} \).
   Prove or disprove the following statement: $\exists x \in \mathbb{Z}, 1 < x < 2.$
3. Consider the set of real numbers \( \mathbb{R} \).
   Analyze the truth value of the following statement: $\forall x \in \mathbb{R}, x^2 \geq x$.
4. Consider the set of positive integers \( \mathbb{P} \).
   Evaluate the truth of the following statement: $\exists x \in \mathbb{P}, x > 100.$