Math Problems Vault

Problem 1.2.1

(a)(b)

Let $f: X \rightarrow Y$ and $A, B \subset X$ . Prove that:

?

(a)

$f(A) \backslash f(B) \subset f(A \backslash B)$ .

(b)

inclusion in (a) can be strict.

Problem 1.2.2

(a)(b)

Let $f: X \rightarrow Y$ , let $A \subset X$ , and $B \subset Y$ . Prove that:

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(a)

If $f$ is surjective then $f\left(f^{-1}(B)\right)=B$ .

(b)

If $f$ is injective then $f^{-1}(f(A))=A$ .

Problem 1.2.3

(a)(b)(c)

Let $f: X \rightarrow Y$ and $A, B \subset X$ . Prove that:

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(a)

$f$ is injective if and only if $f(A \cap B)=f(A) \cap f(B)$ for all $A, B$ .

(b)

$f$ is surjective if and only if $f(A)^{C} \subset f\left(A^{C}\right)$ for all $A$ .

(c)

$f(A)^{C}=f\left(A^{C}\right)$ for all $A$ if and only $f$ is bijective.

Problem 1.2.4

(a)(b)(c)

Let $f: X \rightarrow Y$ and $A, B \subset Y$ . Prove that

?

(a)

$A \subset B$ implies that $f^{-1}(A) \subset f^{-1}(B)$ .

(b)

$f^{-1}(A \cup B)=f^{-1}(A) \cup f^{-1}(B)$ .

(c)

$f^{-1}\left(B^{C}\right)=\left(f^{-1}(B)\right)^{C}$ .

Problem 1.2.5

Let $f: X \rightarrow Y, A \subset X$ , and $B \subset Y$ . Prove that $f\left(A \cap f^{-1}(B)\right)= f(A) \cap B$ .

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Problem 1.2.6

(a)(b)

Let $f: X \rightarrow Y$ , let $X_{0} \subset X$ , let $f_{0}$ be the restriction of $f$ to $X_{0}$ , and let $E \subset Y$ . Prove that:

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(a)

$f_{0}^{-1}(E)=f^{-1}(E) \cap X_{0}$ .

(b)

$f^{-1}(E)=f_{0}^{-1}(E) \cup \left(f^{-1}(E) \cap \left(X \backslash X_{0}\right)\right)$ .

Problem 1.2.7

(a)(b)(c)(d)

Let $f: X \rightarrow Y$ , let $\left\{ A_{i}\right\}_{i \in I}$ be a collection of subsets of $X$ , and let $\left\{ B_{i}\right\}_{i \in I}$ be a collection of subsets of $Y$ . Prove that:

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(a)

$f^{-1}\left(\cap_{i \in I} B_{i}\right)=\cap_{i \in I} f^{-1}\left(B_{i}\right)$ .

(b)

$f\left(\cap_{i \in I} A_{i}\right) \subset \cap_{i \in I} f\left(A_{i}\right)$ .

(c)

The equality holds in (b) if and only if $f$ is injective.

(d)

$f^{-1}\left(\cup_{i \in I} B_{i}\right)=\cup_{i \in I} f^{-1}\left(B_{i}\right)$ .

Problem 1.2.8

(a)(b)

Let $\left\{ A_{i}\right\}_{i \in I}$ and $\left\{ B_{i}\right\}_{i \in I}$ be subsets of $X$ . Prove that

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(a)

$\left(\cup_{i \in I} A_{i}\right) \backslash \left(\cup_{i \in I} B_{i}\right) \subset \cup_{i \in I}\left(A_{i} \backslash B_{i}\right)$ .

(b)

Inclusion in (a) can be strict.

Problem 1.2.9

Let $f: X \rightarrow \mathbb {R}$ and let $A, B \subset X$ . Prove that:

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(a)

$\chi_{A \cap B}=\chi_{A} \chi_{B}$ .

(b)

$\chi_{A \cup B}=\chi_{A}+\chi_{B}-\chi_{A} \chi_{B}$ .

(c)

$\chi_{A \backslash B}=\chi_{A}-\chi_{A} \chi_{B}$ .

(d)

$\chi_{A} \circ f=\chi_{f^{-1}(A)}$ .

(e)

$\left|\chi_{A}-\chi_{B}\right|=\chi_{A \triangle B}$ .