Show that the following pairs of sets have the same cardinality by giving explicit bijections between them

Question

Show that the following pairs of sets have the same cardinality by giving explicit bijections between them:

(a) $M_{2×2}(\mathbb C)$ and $\mathbb R$8

(b) $(0,1)$ and $(−1,+\infty)$

(c) The sets $\{z\in\mathbb C \mid 0<|z|<1,\ 0<\arg(z)<\frac{\pi}4\}\quad\text{ and }\\\{z\in\mathbb C \mid 0<|z|<2,\ \Re(z)<0,\ \Im(z)>0\}$

How would I find the bijections between these sets that have to same cardinality? Thanks.

Answer 1

(a) $M_{2x2}(\mathbb C)\equiv \mathbb C^4\equiv\mathbb R^8$ and $\mathbb R^n$ equipotent to $\mathbb R$ but explicit bijection is not straightforward.

(b) $(0,1)\overset{\frac 1x}{\longmapsto}(1,+\infty)\overset{x-2}{\longmapsto}(-1,+\infty)$

(c) $\{0<|z|<1,0<\theta<\frac{\pi}4\}\overset{z^2}{\longmapsto}\{0<|z|<1,0<\theta<\frac{\pi}2\}\overset{iz}{\longmapsto}\{0<|z|<1,\frac{\pi}2<\theta<\pi\}\overset{2z}{\longmapsto}\{0<|z|<2,\Re(z)<0,\Im(z)>0\}$

Answer 2

For $(a)$, I reckon what's fundamental is that you find a bijection between $\mathbb{R}$ and $\mathbb{R}^2$. Hint: try to use digits of the decimal expansion!

For $(b)$, you can try to modify functions like $\tan(x)$ or $\frac1x$.

For $(c)$, try to draw the sets. The bijection should be pretty obvious once you've pictured them.

Answer 3

(B) set (0'1) have cardinality C and Also (1, infinite) cardinality C Using result same cardinality of the set equivalent Impiles (0'1)~(1'infinite) Implis that is bijective Define map: F(x)=1/x