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jmazurek
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Need help... If f(f(x))=x then prove f is a bijection.
Need help... If f(f(x))=x then prove f is a bijection.
Oh, sorry I meant [0,1). Is there something lacking in the proof? And also, can the reals in [0,1) be put into 1 to 1 correspondence with the rest of the Reals?matt grime said:you mean the interval [0,1) and you appear to have shown surjectivity, or you would if you phrased it as: let r be any real number, r=>0, then there is an x in [0,1) such that [tex]\frac{x}{1-x}=r[/tex]
To prove that a function f is a bijection, you need to show that it is both injective (one-to-one) and surjective (onto). This means that every element in the domain of f maps to a unique element in the range, and that every element in the range has at least one pre-image in the domain.
A bijection is significant because it establishes a one-to-one correspondence between the domain and range of a function. This means that every element in the domain has a unique mapping to an element in the range, and vice versa. This property is useful in many areas of mathematics, including set theory, graph theory, and group theory.
No, a function must be both injective and surjective to be considered a bijection. If a function is injective but not surjective, it is called a one-to-one function. If a function is surjective but not injective, it is called an onto function. Only when a function satisfies both properties is it considered a bijection.
There are several methods for proving that a function is a bijection. One common method is to use the definition of injectivity and surjectivity to show that the function satisfies both properties. Another method is to use a proof by contradiction, assuming that the function is not a bijection and then showing that this leads to a contradiction. Additionally, you can use the properties of inverse functions to prove that a function is a bijection.
Yes, a function can be a bijection even if its domain and range are infinite. For example, the function f(x) = 2x is a bijection from the set of natural numbers to itself, even though both sets are infinite. This is because every natural number has a unique double, and every even natural number has a unique half.