1. ## Very interesting question about sets

A = (a,b)
B = [a,b)
C = [a,b]
D = (a,b]

, a,b are real numbers

Which one of these sets is the biggest?

(Proof is not necessary, I only need to know the correct answer)

2. Originally Posted by Kiki
A = (a,b)
B = [a,b)
C = [a,b]
D = (a,b]

, a,b are real numbers

Which one of these sets is the biggest?

(Proof is not necessary, I only need to know the correct answer)

Bad question: there's no biggest nor smallest. All of them have the same cardinality (assuming, of course, $\displaystyle a<b$...)

Tonio

3. Originally Posted by tonio
Bad question: there's no biggest nor smallest. All of them have the same cardinality (assuming, of course, $\displaystyle a<b$...)

Tonio
Ok, thank you.
Another question:
I have prooved that a function converges in all these 4 sets.
I would just like to use the "biggest" of them to show that it converges in the a larger area (like trying to include all the 4 of them, in a sentence) and I am wondering which one to use.
Maybe the set "C" ?

(assuming , $\displaystyle a<b$...)

4. Originally Posted by Kiki
A = (a,b)
B = [a,b)
C = [a,b]
D = (a,b]

, a,b are real numbers
Which one of these sets is the biggest?
Here is a hint. Define $\displaystyle f:[0,1]\to (0,1)$ by $\displaystyle f(0)=\frac{1}{2},~ f(1)=\frac{1}{3}, f(1/n)=\frac{1}{n+1}, n\ge3,~x\text{ otherwise}$.
It is easy to show that $\displaystyle f$ is a bijection.
Thus we see that $\displaystyle [0,1]~\&~(0,1)$ are ‘same size’.

5. Originally Posted by Kiki
I have prooved that a function converges in all these 4 sets. I would just like to use the "biggest" of them to show that it converges in the a larger area (like trying to include all the 4 of them, in a sentence) and I am wondering which one to use.
Use set $\displaystyle C$ because each of the others are subsets.

6. Originally Posted by Plato
Use set $\displaystyle C$ because each of the others are subsets.
Ok thank you, my question is answered now.

7. But it sounds weird to me that A,B,D are subsets of C, but C is not "bigger" than them.

(My major is not mathematics so I havent studied pure topology)

8. Originally Posted by Kiki
But it sounds weird to me that A,B,D are subsets of C, but C is not "bigger" than them.
In this case what matters is being the supper-set.
That is, if I understand what you mean by “a function converges”.
For example: If $\displaystyle f$ is continuous on $\displaystyle [a,b]$ then it is continuous on $\displaystyle (a,b)$.
"Bigger than" has not meaning in this context.

9. Originally Posted by Plato
what you mean by “a function converges”.
btw I wanted to say "a series of function converges", sorry

Originally Posted by Plato
In this case what matters is being the supper-set.
That is, if I understand what you mean by “a function converges”.
For example: If $\displaystyle f$ is continuous on $\displaystyle [a,b]$ then it is continuous on $\displaystyle (a,b)$.
"Bigger than" has not meaning in this context.
Now its pretty clear, thank you for helping me