Solved: Discrete Functions: One-to-One, Onto Properties

In summary, the conversation discusses the concepts of one-to-one and onto functions, and their relationship to the composition of functions. It is stated that if both f and g are one-to-one, then their composition g(f) is also one-to-one. Similarly, if both f and g are onto, then their composition g(f) is also onto. The attempt at a solution involves using the definitions of one-to-one and onto functions to prove the given statements.
  • #1
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Homework Statement



Suppose A, B, C are sets and [URL]http://latex.codecogs.com/gif.latex?f:A\to%20B,%20\text{%20and,%20}%20g:B\to%20C[/URL]

If f and g are one-to-one so is [PLAIN]http://latex.codecogs.com/gif.latex?g\circ%20f .[/URL]

If f and g are onto, so is [URL]http://latex.codecogs.com/gif.latex?g\circ%20f[/URL]


Homework Equations



One-to-one: [URL]http://latex.codecogs.com/gif.latex?(x,b),(y,b)\epsilon%20f%20\text{%20we%20must%20have%20}%20x=y[/URL]

Onto: Let [URL]http://latex.codecogs.com/gif.latex?f:A\to%20B,[/URL] we say f is onto B provided for every [URL]http://latex.codecogs.com/gif.latex?b\epsilon%20B[/URL] there is an [URL]http://latex.codecogs.com/gif.latex?a\epsilon%20A[/URL] so that [URL]http://latex.codecogs.com/gif.latex?f(a)=b[/URL]

The Attempt at a Solution



So.. I'm sort of lost, I've had this problem in the past, and did not understand it at all, and now its come back to haunt me. I know that:

[URL]http://latex.codecogs.com/gif.latex?im%20f%20=%20B[/URL]
[URL]http://latex.codecogs.com/gif.latex?dom%20f%20=%20A[/URL]
[URL]http://latex.codecogs.com/gif.latex?im%20g%20=%20C[/URL]
[URL]http://latex.codecogs.com/gif.latex?dom%20g%20=%20B[/URL]

But from there I'm sort of lost. Help would be much appreciated!
 
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  • #2
Start with g(f(x))=g(f(y)), and show x=y.
 
  • #3
Call fx=a, fy=b , since g is one to one, what can you can about g(a)=g(b)
 

Related to Solved: Discrete Functions: One-to-One, Onto Properties

1. What is a discrete function?

A discrete function is a type of mathematical function in which the input values are separate and distinct, rather than continuous. This means that the function can only take on specific, isolated values instead of a range of values.

2. What does it mean for a function to be one-to-one?

A one-to-one function is a type of discrete function in which each input value (x) corresponds to a unique output value (y). In other words, each x-value has only one y-value associated with it. This also means that no two input values can have the same output value.

3. How can you determine if a function is one-to-one?

To determine if a function is one-to-one, you can use the horizontal line test. This involves drawing horizontal lines across the function and seeing if they intersect the graph at more than one point. If a horizontal line intersects the graph at more than one point, then the function is not one-to-one.

4. What is the onto property of a function?

The onto property, also known as surjectivity, means that for every output value (y), there is at least one input value (x) that produces that output. In other words, the range of the function must equal the codomain.

5. Can a function be both one-to-one and onto?

Yes, a function can be both one-to-one and onto. This type of function is called a bijection. It means that each input value corresponds to a unique output value and every output value has at least one corresponding input value.

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