Math 3110 Spring 2010 -- Homework Problem Set #1
Due: Friday, January 22nd
- Negate the following statement: "There exists an even integer \( \ell \) such that
for all rational numbers \( r \), \( \;\; r\ell \) is an integer."
Do not try to prove or disprove the statement or its negation.
- Let \( A = \{ m \in \mathbb{Z} \,|\, \exists k \in \mathbb{Z} \mbox{ s.t. } m = 8k+6 \} \)
and let \( B = \{ m \in \mathbb{Z} \,|\, \exists k \in \mathbb{Z} \mbox{ s.t. } m = 4k-2 \} \).
- Show that \( A \subseteq B \).
- Is \( A = B \)?
Don't just answer "Yes" or "No", justify your answer with a proof!
- Let \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) be the function defined by...
\[ f(n) = \left\{ \begin{array}{lc} \frac{n}{2} & n \mbox{ is even} \\
n-3 & n \mbox{ is odd} \end{array} \right. \]
Is \( f \) one-to-one? onto? both? neither?
Prove your answer. If \( f \) is one-to-one, prove it. If \( f \) isn't, show that it
fails to be one-to-one. If \( f \) is onto, prove it. If \( f \) isn't, show that it fails
to be onto.
- Let \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) be the function defined by...
\[ f(n) = \left\{ \begin{array}{lc} n+1 & n \mbox{ is even} \\
4n & n \mbox{ is odd} \end{array} \right. \]
Is \( f \) one-to-one? onto? both? neither?
Prove your answer. If \( f \) is one-to-one, prove it. If \( f \) isn't, show that it
fails to be one-to-one. If \( f \) is onto, prove it. If \( f \) isn't, show that it fails
to be onto.
- Let \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) be the function defined by
\( f(x) = x^4 + 2 \), let \( C = \{ -1,0,1,2 \} \) and let \( D = \{ 0, 1, 2, 3, \dots, 18 \} \).
- Compute \( f(\{-2,3\}) \) and \( f(C) \).
- Compute \( f^{-1}(\{ 2 \}) \), \( f^{-1}(\{0\}) \), and \( f^{-1}(D) \).
- Compute \( f(f^{-1}(D)) \) and notice that this does not give back \( D \). This happens because \( f \) is not onto.
- Let \( f : X \rightarrow Y \) be a surjective function (\( f \) is onto) and let \( B \subseteq Y \). Show
that \( f(f^{-1}(B))=B \).