"one-to-one correspondence"(bijection) and the size of two infinite sets I know little about theory, so the following maybe a stupid question .
"one-to-one correspondence"(bijection) is a good method  to judge whether two finite sets have the same size , while a lot of mathematical rules become invalid in the world of infinite , so 
(1).how Cantor make it sure that bijection of two  infinite sets can also ensure the two  infinite sets have the same size(or the same number of elements) ? 
(2). What "one-to-one correspondence"(bijection) is used for in set theory ?

P.S. The "one-to-one correspondence"(bijection) caused many unexpected surprising result with infinite sets  , maybe wrong result from my perspective , like all positive integers and its subset all positive odd numbers have the same cardinality, the same size(or the same number of elements), however, lots of people think  the number of positive integers should be twice of  its subset all positive even numbers before accepted Cantor's conclusion about this.  Counter-intuitive result caused when comparing the size of two infinite sets using bijection cannot assert comparing the size of two infinite sets using bijection is wrong , but I think if we find another standard to compare the size of two infinite sets and also make the result  useful and accord with our intuition, that would be much better .
 A: First you ask

how Cantor make it sure that bijection of two infinite sets can also
ensure the two infinite sets have the same size?

The answer, which you seem to understand, is that he didn't "make it sure", he made that the definition of "have the same size".
Then you note, correctly, that this definition leads to many counterintuitive results, adding

I think if we find another standard to compare the size of two
infinite sets and also make the result useful and accord with our
intuition, that would be much better .

Perhaps. But mathematicians haven't succeeded in finding another definition that's as generally useful. They've taken an alternative path, refining their intuition about "same size" so as to give results they can prove from Cantor's definition.
You might want to read about some of the controversy surrounding Cantor's work. Some mathematicians at the time tried to do what you suggest. David Hilbert championed Cantor:

No one shall expel us from the Paradise that Cantor has created. (https://en.wikiquote.org/wiki/David_Hilbert)
Cantor's paradise is an expression used by David Hilbert (1926, page
170) in describing set theory and infinite cardinal numbers developed
by Georg Cantor. The context of Hilbert's comment was his opposition
to what he saw as L. E. J. Brouwer's reductive attempts to
circumscribe what kind of mathematics is acceptable; see
Brouwer–Hilbert controversy.

(From https://en.wikipedia.org/wiki/Cantor%27s_paradise)
A: The existence or nonexistence of injections and bijections is very important and useful in mathematics. I don't think you have a problem with our use of injections and bijections. What you don't like is the use of the word "size" in talking about cardinalities of sets. Maybe you think it would be better to speak of sets having the same "weight" or "capacity" or "beauty" if there is a bijection between them. To this I have two answers.
(1) Be happy, "size" is not an official technical term (except, to be honest, in certain contexts such as graph theory); the technical term is "cardinality".
(2) Yes, mathematicians do think (informally) of uncountable sets as being somehow "bigger" than countable sets, and of all countable sets (the integers, the prime numbers, the algebraic numbers) as being "the same size". You find this counterintuitive. But don't you think the people who actually use injections, bijections, and cardinals in their work are in a better position to decide what kind of imagery and metaphor is helpful to them?
A: The definition of equal cardinality of sets is that there exists a bijection between them. 
The intuition to this definition comes from finite sets; suppose you have a set of boys and a set of girls. How can you tell if there are as many boys as girls? Try to match each boy to one unique girl. That's exactly the definition of a bijection - a matching, i.e. a one-to-one and onto function between two sets. 
When considering an infinite set, one needs a formal definition of cardinality (i.e. size). We cannot use the same intuitive definition as in finite sets ("the number of elements in the set"), since the set is infinite. Instead, we use the formal definition of quality of cardinality in order to, in some way, partition the different sets to equivalence classes. 
As for your second question, bijections have many uses in mathematics. I cannot really provide an answer that consists of all of their uses. What is your background in math? Perhaps I can provide an better answer that relates to other things you know. 
