[a,b) and (a,b) are equinumerous

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In summary, there is a discussion about finding a bijective function from the interval $[a,b)$ to $(a,b)$, where the inclusion mapping is not enough as it is not surjective. The conversation suggests considering the case of $a=0$ and $b=1$ and constructing a bijection using a piecewise-defined function. The suggested function maps $0$ to $1/2$ and $1/n$ to $1/(n+1)$ for $n\ge2$, while all other numbers are mapped to themselves.
  • #1
evinda
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Hi! (Smile)

I want to show that the interval $[a,b)$ is equinumerous with this one: $(a,b)$.
How could we find a bijective function from $[a,b)$ to $(a,b)$? (Thinking)
 
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  • #2
Well, $(a,b) \subset [a,b)$. Wouldn't this solve everything?
 
  • #3
Fantini said:
Well, $(a,b) \subset [a,b)$. Wouldn't this solve everything?

Not exactly. The inclusion mapping $i : (a,b)\to [a,b)$, although injective, is not surjective: there is no $x \in (a,b)$ such that $i(x) = a$.

Evinda, I suggest first considering the case $a = 0$ and $b = 1$. Construct a bijection $f : [0,1) \to (0,1)$ using a piecewise-defined function.
 
  • #4
Euge said:
Evinda, I suggest first considering the case $a = 0$ and $b = 1$. Construct a bijection $f : [0,1) \to (0,1)$ using a piecewise-defined function.

What piecewise-defined function could we use? :confused:
 
  • #5
Map $0$ to $1/2$ and $1/n$ to $1/(n+1)$ for $n\ge2$. All other numbers are mapped to themselves.
 

FAQ: [a,b) and (a,b) are equinumerous

1. What does it mean for two sets to be equinumerous?

Two sets are equinumerous if they have the same number of elements. In other words, there is a one-to-one correspondence between the elements of the two sets. This means that each element in one set can be paired up with a unique element in the other set.

2. What is the difference between (a,b) and [a,b)?

The notation (a,b) represents an open interval, which includes all real numbers between a and b, but does not include a and b themselves. On the other hand, [a,b) represents a half-open interval, which includes all real numbers between a and b, including a, but not b.

3. How can you prove that (a,b) and [a,b) are equinumerous?

To prove that two sets are equinumerous, you need to establish a one-to-one correspondence between the elements of the two sets. In the case of (a,b) and [a,b), you can establish this correspondence by mapping every element x in (a,b) to x-1 in [a,b). This mapping satisfies the definition of a one-to-one correspondence, proving that the two sets are equinumerous.

4. Can (a,b) and [a,b) be equinumerous if a and b are not real numbers?

No, in order for two sets to be equinumerous, they must have the same number of elements. If a and b are not real numbers, then the sets (a,b) and [a,b) are not well-defined and do not have a specific number of elements, making it impossible for them to be equinumerous.

5. How does the concept of equinumerosity relate to cardinality?

Equinumerosity is a fundamental concept in the study of cardinality, which is the measure of the size or number of elements in a set. Two sets are equinumerous if they have the same cardinality, meaning they have the same number of elements. This is one way to compare the sizes of different sets and is an important concept in mathematical set theory.

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