Can someone teach me this problem of sets.

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In summary: Thanks for the help!In summary, the function G is 1-1 if and only if the function f is 1-1, and G is onto if and only if the function f is onto.
  • #1
S&S
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Let A and B be sets and Let f:A---->B be a function. Define a function G: Power set (B)------->Power (A) by declaring that, for C is part of B or equal to B:
G(C)={x is element of A: f(x) is element of C}
Show that G is 1-1 if and only if f is onto. :cry:

Can anyone teach me this problem? Hopefully I can get some feelings about set theory. I just don't feel right.

Thank you so much for your good tips.
 
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  • #2
When you look at a problem like this and it means nothing to you, look at simple examples. Suppose A is the set {1,2,3} and B is {a,b}.
1) First example. Define f by f(1)= a, f(2)= a, f(3)= a (f is neither 1-1 nor onto). The power set of B is {{}, {a}, {b}, {a,b}}. G({})= {}, of course, because every member of A is mapped into something. G({a})= {1, 2, 3} because every member of A is mapped into a. G({b})= {} because no member of A is mapped into b. G({a,b})= {1,2,3} because every member of A is mapped into a which is in {a,b}. This is not 1-1 because two different sets, {} and {a} are mapped into {} and, also, two different sets, {a} and {a,b} are mapped into {1, 2, 3}. It is not onto because there is no set which is mapped into, for example, {1}, a set in the powerset of A.
2) Second example. g(1)= a, g(2)= a, g(3)= b (g is onto but not 1-1).
G({})= {}, G({a})= {1,2}, G({b})= {3}, G({a,b})= {1, 2, 3}. Now G is 1-1 since no two members of the power set of B are mapped into the same thing. It is NOT onto because no set is mapped into, for example, {1, 2}.

We can't give an example with these two sets of a 1-1 function because A has too many elements. Take A= {1,2}, B= {a,b,c} and define f(1)= a, f(2)= b. Now f is 1-1 but not onto. G({})= {}, G({a})= {1}, G({b})= {2}, G({c})= {}, G({a,b})= {1,2}, G({a,c})= {1}, G({b,c})= {2}, G({a,b,c})= {1,2}. G is onto but not 1-1.

That's interesting. Perhaps we can simplify the problem by doing it as two separate parts:
(1) the function G is 1-1 if and only if f is 1-1 and
(2) the function G is onto if and only if f is onto.
 
  • #3
just say cheers

I'm getting better.
 

What is a set?

A set is a collection of distinct objects or elements that are grouped together for a specific purpose or based on a certain criteria.

Why is understanding sets important in science?

Sets are important in science because they allow us to organize and categorize data, observations, and phenomena. They also help us to identify patterns and relationships between different elements.

How can someone teach me about sets?

One way to learn about sets is to study the basic principles and properties of sets, such as intersection, union, and complement. You can also learn about different types of sets, such as finite and infinite sets, and how to use different notation and symbols to represent sets.

What are some real-life applications of sets?

Sets are used in various fields of science, such as biology, chemistry, and physics, to classify and organize data. They are also used in computer science and data analysis to sort and filter information. In everyday life, sets can be used to group items, such as books on a bookshelf or ingredients in a recipe.

Are there any online resources for learning about sets?

Yes, there are many online resources available for learning about sets, including tutorials, videos, and interactive exercises. Some recommended resources include Khan Academy, Math is Fun, and Brilliant.org.

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