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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