Classifying Functions from {1,2,3} to {1,2} and Finding Right Inverses

In summary, the conversation discusses identifying functions from one set to another using arrow diagrams. The concepts of injective, surjective, and bijective functions are also mentioned, with examples provided for each. The conversation concludes with a reminder of the definitions of these types of functions.
  • #1
L²Cc
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Homework Statement


List all the functions from {1,2,3} to {1,2} representing each function as an arrow
diagram. Which of these functions are (a) injective, (b) surjective, (c) bijective? For
each surjective function write down a right inverse.


Homework Equations



The Attempt at a Solution


The only one I can think of is,
f(x) = x - 1, where X is all real numbers,
and in the directions it specifically asks for functionS...
 
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  • #2
You're think in terms of formulas of functions, when that isn't the direction you should go. In fact, function formulas have no role to play in this problem. Recall that a function is simply a mapping of members in one set to those of another set such that no member of the first set gets mapped to multiple members of the second set.

Look at your problem description again. You are supposed to be drawing arrows from the members of the first set, {1, 2, 3}, to the second set, {1, 2}. Look in your textbook or notes for the arrow diagrams that your problem refers to.
 
  • #3
oh right right!
in other words,
f(1) = 1
f(2) = 2
f(3) = 2
functions f(2) and f(3) would have been injections had a and b in the following f(a) = f(b)were equal which is not the case here? Surjections have have the property that for every y in the codomain there is an x in the domain such that ƒ(x) = y, f(1) and f(2) surjections? Not f(3) because it maps to the same element in the range as f(2) ?

Thank you.
 
  • #4
Your f is surjective since every element in the codomain are in the image of f. It is not injective since it f(2)=f(3), and therefore is not a bijection.

Recall that a function is injective or one to one if all elements in the domain map to unique elements of the codomain. A function is surjective or onto if all elements from the codomain are covered. An a bijection is a function that is both an injection and a surjection.
 
Last edited:
  • #5
Makes sense. Thank you!
 

What is a set?

A set is a collection of distinct and well-defined objects. These objects can be anything, such as numbers, letters, or even other sets.

What is a relation?

A relation is a connection or association between two sets. It describes how the elements of one set are related to the elements of another set.

What is a function?

A function is a special type of relation where each input from one set is mapped to exactly one output in another set. In other words, each element in the domain has one and only one corresponding element in the range.

What is the difference between a one-to-one function and an onto function?

A one-to-one function is a function where each element in the domain is uniquely mapped to an element in the range, meaning no two different inputs can have the same output. An onto function is a function where every element in the range has at least one corresponding element in the domain, meaning there are no "leftover" elements in the range.

How are sets, relations, and functions related to each other?

Sets, relations, and functions are all fundamental concepts in mathematics. Sets provide the foundation for relations and functions, which describe how elements in a set are related or connected. Functions are a special type of relation that is used to model real-world situations and solve problems in various fields of study.

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