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4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then |AN B| 2 3. 5. Prove the following theorem from class: Theorem 1. Suppose that X and Y are nonempty finite sets with |X| < |Y| and f: X → Y is a function. Then f is not a surjection.

4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then |AN B| 2 3. 5. Prove the following theorem from class: Theorem 1. Suppose that X and Y are nonempty finite sets with |X| < |Y| and f: X → Y is a function. Then f is not a surjection.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q6 Mathmetical reasoning. Please with full explanation and steps. I struggle with this topic very much. Explanation and typed answers will be appreciated. Thanks！
1. **Let \( f: X \to Y \) be a function.** (a) Prove that \( \overset{\leftarrow}{f}(B_1 \cap B_2) = \overset{\leftarrow}{f}(B_1) \cap \overset{\leftarrow}{f}(B_2) \) for any \( B_1, B_2 \in \mathcal{P}(Y) \). (b) Prove that \( \overset{\rightarrow}{f}(A_1 \cap A_2) \subseteq \overset{\rightarrow}{f}(A_1) \cap \overset{\rightarrow}{f}(A_2) \) for any \( A_1, A_2 \in \mathcal{P}(X) \). Give an example of a specific function \( f \) and sets \( A_1, A_2 \) such that \( \overset{\rightarrow}{f}(A_1) \cap \overset{\rightarrow}{f}(A_2) \nsubseteq \overset{\rightarrow}{f}(A_1 \cap A_2) \). 2. **Let \( f: X \to Y \) be a function.** (a) Prove that \( f \) is injective \( \iff \overset{\rightarrow}{f} \) is injective \( \iff \overset{\leftarrow}{f} \) is surjective. (b) Prove that \( f \) is surjective \( \iff \overset{\rightarrow}{f} \) is surjective \( \iff \overset{\leftarrow}{f} \) is injective. _(Hint: when proving statements of the form \( P \iff Q \iff R \), it is often easier to prove \( P \implies Q \implies R \) than to prove \( P \iff Q \) and \( Q \iff R \)). 3. **Prove Corollary 10.2.2 from class: for any \( n \in \mathbb{Z}_+ \), if \( X_1, \ldots, X_n \) are finite sets which are pairwise disjoint, then** \[ \left| \bigcup_{i=1}^n X_i \right
Transcribed Image Text:1. **Let \( f: X \to Y \) be a function.** (a) Prove that \( \overset{\leftarrow}{f}(B_1 \cap B_2) = \overset{\leftarrow}{f}(B_1) \cap \overset{\leftarrow}{f}(B_2) \) for any \( B_1, B_2 \in \mathcal{P}(Y) \). (b) Prove that \( \overset{\rightarrow}{f}(A_1 \cap A_2) \subseteq \overset{\rightarrow}{f}(A_1) \cap \overset{\rightarrow}{f}(A_2) \) for any \( A_1, A_2 \in \mathcal{P}(X) \). Give an example of a specific function \( f \) and sets \( A_1, A_2 \) such that \( \overset{\rightarrow}{f}(A_1) \cap \overset{\rightarrow}{f}(A_2) \nsubseteq \overset{\rightarrow}{f}(A_1 \cap A_2) \). 2. **Let \( f: X \to Y \) be a function.** (a) Prove that \( f \) is injective \( \iff \overset{\rightarrow}{f} \) is injective \( \iff \overset{\leftarrow}{f} \) is surjective. (b) Prove that \( f \) is surjective \( \iff \overset{\rightarrow}{f} \) is surjective \( \iff \overset{\leftarrow}{f} \) is injective. _(Hint: when proving statements of the form \( P \iff Q \iff R \), it is often easier to prove \( P \implies Q \implies R \) than to prove \( P \iff Q \) and \( Q \iff R \)). 3. **Prove Corollary 10.2.2 from class: for any \( n \in \mathbb{Z}_+ \), if \( X_1, \ldots, X_n \) are finite sets which are pairwise disjoint, then** \[ \left| \bigcup_{i=1}^n X_i \right
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