Show that the following pairs of sets have the same cardinality.

(a) Integers divisible by 3, and the even positive integers.

(b) $$\Bbb {R}$$, and the interval $$(0,\infty)$$.

(c) The interval $$[0, 2)$$, and the set [5,6)∪[7,8).

(d) The intervals $$(-\infty, -1)$$ and $$(-1, 0)$$.

I know that cardinality means that there is a bijection between the two sets, and that means there is a surjection and injection. For the first one I think you can simply make a function that is a bijection and prove that it is an injection and surjection. I am not sure how to do the last three.

Sorry for the poor formatting in advance and if you need more information about the question I am more than happy to provide it.

Thank you.

• ok thank you, fixed it – smith Nov 27 '18 at 2:47
• Do you mean the set $[5,6)\cup [7,8)$? – TonyK Nov 27 '18 at 2:50
• yes, it has been fixed – smith Nov 27 '18 at 2:51

Hints:

(b) Map $$x \in \mathbb{R}$$ by $$x \to e^x$$

(c) Map $$x \in [0, 2)$$ by $$x \to x + 5$$ if $$x \in [0,1)$$ and $$x \to x + 6$$ if $$x \in [1,2)$$

(d) Map $$x \in (-\infty,-1)$$ by $$x \to \frac{1}{x}$$

Show that these are all bijections.

(b) $$\mathbb{R}$$, and the interval (0,∞).

Consider $$f : \mathbb{R}\rightarrow (0,\infty)$$ defined by $$f(x)=e^x.$$

(c) The interval [0,2), and the set [5,6) or [7,8).

Consider $$g: [0,2)\rightarrow [5,6)$$ defined by $$g(x)=5+\frac{x}{2}$$ or $$g'(x)=7+\frac{x}{2}$$ if codomain is $$[7,8)$$.

(d) The intervals (−∞,−1) and (−1,0).

Consider $$h: (-1,0)\rightarrow (-\infty,-1)$$ defined by $$h(x)=\tan(\frac{\pi}{2}x)-1.$$

a) Let $$A$$ be a set of integers divisible by 3 and $$C$$ a set of positive even integers. You can take a composition of following functions. First $$f:A\to \mathbb{Z}$$ defined with $$n\longmapsto {n\over 3}$$

and then $$g:\mathbb{Z}\to C$$ defined by:

$$g(n) \begin{cases} = 4n+2 & \text{if } n\geq 0, \\ = 4|n| & \text{if } n < 0. \\ \end{cases}$$

Then let $$g\circ f$$ will do the job.

The bijections, reading left to right, can be chosen as:

(a) $$|4x/3|+2[x\ge 0]$$ (b) $$x+\sqrt{1+x^2}$$ (c) $$x+5+\lfloor x\rfloor$$ (d) $$1/x$$.