# Problem Set 1

1. Easy
Let $$A\subseteq S$$ and $$B\subseteq S$$. Prove each of the following statements
1. $$A\subseteq B$$ if and only if $$A\cup B = B$$
2. $$A^{c}\subseteq B$$ if and only if $$A\cup B = S$$
3. $$A\subseteq B$$ if and only if $$B^{c}\subseteq A^{c}$$
4. $$A\subseteq B^{c}$$ if and only if $$A\cap B = \emptyset$$

2. Easy
Let $$A$$, $$B$$, and $$C$$ be subsets of $$S$$. Show that if $$A\subseteq B$$ and $$B\subseteq C$$, then $$A\subseteq C$$.

3. Easy
Let $$f:A\to B$$ be a map of sets, and let $$\left\{X_{i}\right\}_{i\in I}$$ be an indexed collection of subsets of $$A$$.
1. Prove that $$f\left(\bigcup_{i\in I} X_{i}\right) = \bigcup_{i\in I} f(X_{i})$$
2. Prove that $$f\left(\bigcap_{i\in I} X_{i}\right) \subset \bigcap_{i\in I} f(X_{i})$$
3. When does equality of sets hold in the above part?

4. Easy
Consider the map $$f: \mathbb R \to \mathbb R, x \mapsto \sin(x)$$.
1. Find the image of the set $$[0,\pi]$$; that is, determine $$I = f([0,\pi])$$.
2. Let $$I$$ be as in the previous question. Determine the preimage $$f^{-1}(I)$$.

5. Easy
Consider the function $$F: \mathbb R^2 \to \mathbb R, (x,y) \mapsto y - x^2$$.
1. Graph the set $$F^{-1}(0)$$ in $$\mathbb R^2$$.
2. More generally, if $$c \in \mathbb R$$ is any real number, what does the set $$F^{-1}(c)$$ look like?

6. Medium
Let $$I$$ be an index for a collection of subsets $$A_{i}\subseteq S$$, $$i \in I$$. Show that for every $$k \in I$$, $$\bigcap_{i\in I}A_{i}\subseteq A_{k}$$

7. Medium
Give an example of a function $$f(x) = \left( f_1(x), f_2(x) \right)$$, such that neither of the component functions $$f_1,f_2$$ are injective, but $$f$$ is injective. Be sure to specify the domain of $$f$$.

8. Hard
Let $$f: A\to B$$ be a function.
1. For every $$X\subseteq A$$, $$X \subseteq f^{-1}(f(X))$$
2. For every $$Y\subseteq B$$, $$Y \supseteq f(f^{-1}(Y))$$
3. If $$f: A\to B$$ is injective, then for every $$X\subseteq A$$ we have $$X = f^{-1}(f(X))$$
4. If $$f:A\to B$$ is surjective, then for every $$Y\subseteq B$$ we have $$Y = f(f^{-1}(Y))$$

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