Exercises - Cardinality and Infinite Sets

  1. Decide if each function described is injective, surjective, bijective, or none of these, and justify your decision. (For parts (e)-(g), note that $2\mathbb{Z}$ represents the set of all even integers.)  

    1. $f:\mathbb{N}\rightarrow\mathbb{N}$ where $f(n)=n+1$

    2. $f:\mathbb{Z}\rightarrow\mathbb{Z}$ where $f(n)=n+1$

    3. $f:\mathbb{N}\rightarrow\mathbb{N}$ where $f(n)=n^2$

    4. $f:\mathbb{Z}\rightarrow\mathbb{Z}$ where $f(n)=n^2$

    5. $f:\mathbb{Z} \rightarrow 2\mathbb{Z}$ where $f(n) = 2n+2$

    6. $f:\mathbb{N} \rightarrow 2\mathbb{Z}$ where $f(n) = 2n+2$

    7. $f:2\mathbb{Z} \rightarrow \mathbb{N}$ where $f(n) = \displaystyle{\frac{|n|}{2}}$

    8. $f:\mathbb{Q} \rightarrow \mathbb{Q}$ where $f(n) = \displaystyle{\frac{1}{x^2+1}}$

  2. Find a bijection between the sets $\{1,2,3,4,\ldots\}$ and $\{7,10,13,16,\ldots\}$.

  3. Find a bijection between the real numbers in the interval $[0,\infty)$ and the real numbers in the interval $(0,\infty)$.

  4. Find a bijection between the real numbers in the interval $[0,1]$ and the real numbers in the interval $(0,1)$.

  5. Describe a bijection between the set of points on a unit square and the set of points on a semi-circle.

  6. Describe a bijection between the set of points making up a line segment and the set of point making up a cube.

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