Proving a function is onto

Thread starter gottfried
Start date Nov 10, 2012
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Function

In summary, being onto means that every element in the range of a function has at least one corresponding input value in the domain. To prove that a function is onto, you must show that every element in the range is mapped to by at least one element in the domain. This can be done by showing that for every output in the range, there exists at least one input that produces that output when plugged into the function. The difference between onto and one-to-one functions is that an onto function maps every element in the range to by at least one element in the domain, while a one-to-one function maps each input to a unique output. A function can be both onto and one-to-one, in which case it is called a bijection

Nov 10, 2012

gottfried

Homework Statement

Let G be a group and define λ_g:G→G to be λ_g(x)=g.x , x[itex]\in[/itex]G.

Show that λ_g is onto and one-to-one.

The Attempt at a Solution

Suppose g.x=g.x' g^-1.g.x=g^-1.g.x' which means x=x'.

How should I show that the function is onto?

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Nov 10, 2012

HallsofIvy

Science Advisor

Homework Helper

If y is any member of G, then [itex]\lambda(g^{-1}y)= y[/itex]

1. What does it mean for a function to be onto?

Being onto, also known as surjective, means that every element in the range of a function has at least one corresponding input value in the domain. In other words, every output of the function is mapped to by at least one input.

2. How can I prove that a function is onto?

To prove that a function is onto, you must show that every element in the range is mapped to by at least one element in the domain. This can be done by showing that for every output in the range, there exists at least one input that produces that output when plugged into the function.

3. What is the difference between onto and one-to-one functions?

An onto function maps every element in the range to by at least one element in the domain, while a one-to-one function maps each input to a unique output. A function can be both onto and one-to-one, in which case it is called a bijection.

4. Can a function be onto if it is not one-to-one?

Yes, a function can be onto even if it is not one-to-one. This means that some elements in the range may be mapped to by multiple elements in the domain. However, all elements in the range must still be mapped to by at least one element in the domain for the function to be onto.

5. How can I disprove that a function is onto?

To disprove that a function is onto, you must provide a counterexample where an element in the range is not mapped to by any element in the domain. This would show that the function is not onto, as there exists at least one output with no corresponding input.