A Surjective function from [0,1]\{1/2} to [0,1]

In summary, the conversation is about a student seeking help with a question in their Discrete Mathematics homework. They are looking for a Surjective function from [0,1]\{1/2} to [0,1] such that if f(a)>f(b), then a>b. The student is not allowed to use arguments involving cardinality. They are given a hint to consider an increasing sequence that converges to 1/2 and to think about the point $\lim_{n\to\infty}f(x_n)$.
  • #1
Ella1
2
0
Hello guys! I'm taking Discrete Mathematics this semester and I got this question in one of my homework tasks.
I've tried thinking about the solution over and over but can't seem to come up with anything..
The question goes like this: Is there a Surjective function from [0,1]\{1/2} to [0,1] such that if f(a)>f(b) that means that a>b?
I must mention that sadly I cannot use any arguments involving cardinality..
Any clue that might help will save my life!p.s Sorry for my poor English :)!
 
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  • #2
Ella said:
Hello guys! I'm taking Discrete Mathematics this semester and I got this question in one of my homework tasks.
I've tried thinking about the solution over and over but can't seem to come up with anything..
The question goes like this: Is there a Surjective function from [0,1]\{1/2} to [0,1] such that if f(a)>f(b) that means that a>b?
Hi Ella, and welcome to MHB!

Suppose that $(x_n)$ is an increasing sequence in $[0,1/2)$, with $\lim_{n\to\infty}x_n = 1/2.$ What can you say about the point $\lim_{n\to\infty}f(x_n)$? Can it be in the range of $f$?
 

FAQ: A Surjective function from [0,1]\{1/2} to [0,1]

What is a surjective function?

A surjective function, also known as a "onto" function, is a type of function in mathematics where every element in the range of the function is mapped to by at least one element in the domain of the function. In simpler terms, it means that every output has at least one input that produces it.

What is the domain and range of a surjective function from [0,1]\{1/2} to [0,1]?

The domain of this function would be the set of all real numbers from 0 to 1, excluding the number 1/2. The range would also be the set of all real numbers from 0 to 1, inclusive.

How is a surjective function different from an injective function?

A surjective function is different from an injective function in that a surjective function can have multiple inputs that produce the same output, while an injective function has a unique input for every output. In other words, a surjective function covers the entire range, while an injective function covers the entire domain.

Can a surjective function have an infinite range?

Yes, a surjective function can have an infinite range. As long as every element in the range is mapped to by at least one element in the domain, the function is considered surjective.

How do you know if a function is surjective?

To determine if a function is surjective, you can use the horizontal line test. This test involves drawing a horizontal line across the range of the function. If the line intersects the graph of the function at least once for every point in the range, then the function is surjective. Alternatively, you can also check if every element in the range has at least one pre-image in the domain.

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