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You've uploaded an image containing several mathematical problems. I'll help you with these questions step-by-step. Here's a brief breakdown of the problems:

1. **Problem 1**: Prove a set operation identity: \( A \cap (B \setminus C) = (A \cap B) \setminus (A \cap C) \).
2. **Problem 2**: Prove that for an injective function \( f: A \to B \), if \( E \subseteq A \), then \( f^{-1}(f(E)) = E \).
3. **Problem 3**: Prove the inequality \( 2n - 3 < 2^{n-2} \) for all \( n \geq 5 \).
4. **Problem 4**: Prove that if \( a \in \mathbb{R} \) and \( a \cdot a = a \), then \( a = 0 \) or \( a = 1 \).
5. **Problem 5**: Prove that for \( 0 \leq a < b \), \( a^2 \leq ab < b^2 \).
6. **Problem 6**: Prove that if \( S \) is bounded above and \( b > 0 \), then for \( bS := \{bs : s \in S \} \), \( \inf(bS) = b \cdot \inf(S) \).
7. **Problem 7**: Prove using the definition of a limit that
   \[
   \lim_{n \to \infty} \frac{2n^2}{3n^2 + 1} = \frac{2}{3}.
   \]

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You've uploaded an image containing several mathematical problems. It covers set theory, functions, inequalities, real numbers, limits, and more.

Explore a set of challenging mathematical problems involving sets, injective functions, inequalities, and limits. Detailed step-by-step solutions provided. Suitable for undergraduate students.

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