Proving Equivalence of Sets - One-to-One and Onto Function

Thread starter MathMajor
Start date Sep 25, 2009
Tags

Equivalence Function Sets

In summary, a one-to-one function in proving equivalence of sets is a type of function where each element in the domain is mapped to a unique element in the codomain. An onto function, also known as a surjective function, is a type of function where every element in the codomain has at least one corresponding element in the domain. When a function is both one-to-one and onto, it is referred to as a bijective function, which is important in proving equivalence of sets because it guarantees that the two sets being compared have the same number of elements. To prove the equivalence of two sets using one-to-one and onto functions, you must find a bijective function between them. A function can be one-to-one but not onto

Sep 25, 2009

MathMajor

Homework Statement

Hi, new to the Physics Forum and desperately need some help with a math analysis problem...

Prove that {x|x>1} and {x|0<x<1} are equivalent sets by writing a function and show that it is one-to-one and onto.

Homework Equations

The Attempt at a Solution

Physics news on Phys.org

Sep 25, 2009

VeeEight

The sets are not equivalent, rather they have the same cardinality.

1. What is the concept of one-to-one function in proving equivalence of sets?

In proving equivalence of sets, a one-to-one function is a type of function where each element in the domain is mapped to a unique element in the codomain. This means that no two elements in the domain are mapped to the same element in the codomain. In other words, each input has only one corresponding output.

2. How is an onto function defined in the context of proving equivalence of sets?

An onto function, also known as a surjective function, is a type of function where every element in the codomain has at least one corresponding element in the domain. This means that for every output, there exists at least one input that maps to it. In other words, the range of the function is equal to the codomain.

3. What is the significance of proving that a function is both one-to-one and onto?

When a function is both one-to-one and onto, it is referred to as a bijective function. This type of function ensures that every element in the domain has a unique corresponding element in the codomain, and every element in the codomain has at least one corresponding element in the domain. This is important in proving equivalence of sets because it guarantees that the two sets being compared have the same number of elements and can therefore be considered equivalent.

4. How do you prove the equivalence of two sets using one-to-one and onto functions?

In order to prove the equivalence of two sets using one-to-one and onto functions, you must show that there exists a bijective function between the two sets. This means that you must find a function that is both one-to-one and onto, and that maps the elements of one set to the elements of the other set. This proves that the two sets have the same number of elements and are therefore equivalent.

5. Can a function be one-to-one but not onto, or vice versa?

Yes, a function can be one-to-one but not onto, or vice versa. For a function to be one-to-one, each element in the domain must have a unique corresponding element in the codomain. However, this does not guarantee that every element in the codomain has a corresponding element in the domain. Similarly, for a function to be onto, every element in the codomain must have at least one corresponding element in the domain, but this does not guarantee that each element in the domain has a unique corresponding element in the codomain.