- #1
algonewbee
Let A and B be finite sets, and let f:A->B be a function. Show that
a)if f is injective, then |A|<=|B|
b)if f is surjective, then |A|>=|B|
a)if f is injective, then |A|<=|B|
b)if f is surjective, then |A|>=|B|
Injectivity and surjectivity are properties of functions in mathematics. A function f:A->B is said to be injective if for every element in the domain A, there exists a unique element in the range B. In other words, no two elements in A map to the same element in B. On the other hand, a function is said to be surjective if every element in the range B has at least one corresponding element in the domain A that maps to it.
To prove that a function is injective, you need to show that for any two elements in the domain A, if they have the same image in the range B, then they must be the same element. This can be done by using a logical argument or a proof by contradiction. To prove surjectivity, you need to show that every element in the range B has at least one pre-image in the domain A.
Proving injectivity and surjectivity of a function is important because it helps us understand the behavior of the function and its relationship between the domain and range. It also allows us to determine if the function has an inverse, which is crucial in many mathematical and scientific applications.
Yes, a function can be both injective and surjective. Such a function is called a bijective function. In this case, every element in the domain A maps to a unique element in the range B, and every element in the range B has a corresponding element in the domain A. Bijective functions are invertible, meaning they have a unique inverse function.
Some common examples of injective functions include the identity function, where every element in the domain maps to itself, and the function f(x)=x^2, where the output is always positive for any input in the domain. Examples of surjective functions include the exponential function f(x)=e^x, where every real number has a corresponding positive output, and the absolute value function, where both positive and negative inputs have the same positive output.