1. ## Equinumerous

Show that the following pairs of sets S and T are equinumerous by finding a specific bijection between the two sets in each pair.

a. S= [0,1] and T = [1,3]

b. S= [0,1] and T = [0, infinity]

2. Originally Posted by GoldendoodleMom
Show that the following pairs of sets S and T are equinumerous by finding a specific bijection between the two sets in each pair.

a. S= [0,1] and T = [1,3]
Think geometrically. Length of $\displaystyle [0,1]$ is $\displaystyle 1$ and length of $\displaystyle [1,3]$ is $\displaystyle 2$. Thus, strech $\displaystyle [0,1]$ to $\displaystyle [0,2]$. Now shift this interval by one to the right this gives $\displaystyle [1,3]$.

Now the streching can be desribed by function $\displaystyle f :[0,1] \to [0,2]$ by $\displaystyle f(x) = 2x$. And the shifting can be described by $\displaystyle g:[0,2]\to [1,3]$ by $\displaystyle g(x) = x+1$. Thus, $\displaystyle g\circ f: [0,1] \to [1,3]$ is what you are looking for. Finally $\displaystyle g(f(x)) = 2x+1$ is the mapping you want.

3. Originally Posted by GoldendoodleMom
Show that the following pairs of sets S and T are equinumerous by finding a specific bijection between the two sets in each pair.

b. S= [0,1] and T = [0, infinity]
Consider $\displaystyle f:[0,1) \to [0, \infty)$ defined by $\displaystyle f(x)=\frac{x}{1-x}$ for $\displaystyle x \in [0,1)$

RonL

4. Originally Posted by CaptainBlack
Consider $\displaystyle f:[0,1) \to [0, \infty)$ defined by $\displaystyle f(x)=\frac{x}{1-x}$ for $\displaystyle x \in [0,1)$

RonL
Also an alternative is to use $\displaystyle \tan$ and/or $\displaystyle \tan^{-1}$ (inverse of $\displaystyle \tan$), these really help too much.
$\displaystyle f:[0,1)\to[0,\infty)$
.........$\displaystyle t\to f(t)=\tan\bigg(\frac{\pi}{2}t\bigg)$