- #1
woundedtiger4
- 188
- 0
As the title says.
Yes sir, exactly.Nugatory said:Are you asking what term we would use to describe a mapping that is neither injective, surjective, nor bijective? Or are you asking for an example of suchba mapping?
Thank you sirWWGD said:To be more precise, as nuuskur pointed out, the function ## f : \mathbb R \rightarrow \mathbb R ## defined by ## f(x)= x^2 ## is neither injective nor surjective; f(x)=f(-x) , and no negative number is the image of any number. You need to clearly state your domain and codomain, otherwise every function is trivially surjective onto its image. If you changed/restricted the domain, OTOH, you can make the same _expression_ ##f(x)=x^2 ## a bijection from the positive Reals to themselves. While long-winded, the point is that you must define your domain and codomain in order to define a function and study its properties unambiguously.
nuuskur said:Yes, thank you WWGD. I apologize for my lazy explanation.
woundedtiger4 said:Thank you sir
A function is considered injective if each input value has a unique output value. A surjective function has every possible output value represented by at least one input value. A bijective function satisfies both of these conditions, meaning each input has a unique output and every output has at least one corresponding input. Therefore, a function that is neither injective, surjective, nor bijective does not meet these criteria.
To determine if a function is injective, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not injective. To determine if a function is surjective, you can check if every possible output value is represented by at least one input value. To determine if a function is bijective, you can use both the horizontal line test and check if every output value has at least one corresponding input value.
An example of a function that is neither injective, surjective, nor bijective is f(x) = x^2. This function is not injective because, for example, both x = 2 and x = -2 result in the output value of 4. It is not surjective because there is no input value that can produce a negative output value. It is also not bijective because it fails to meet both the criteria of injectivity and surjectivity.
Injective, surjective, and bijective functions are important in mathematics because they help us understand the relationship between input and output values. Injective functions can be thought of as one-to-one mappings, surjective functions as onto mappings, and bijective functions as both one-to-one and onto mappings. These concepts are fundamental in many branches of mathematics, including calculus, linear algebra, and abstract algebra, and have practical applications in fields such as computer science and data analysis.
No, a function cannot be both injective and surjective but not bijective. If a function is injective, it means that each input has a unique output. If it is surjective, it means that every possible output value is represented by at least one input value. Therefore, a function that is both injective and surjective must have a one-to-one correspondence between input and output values, making it bijective.