Group Theory Sets and Mappings

In summary, Group Theory is a branch of mathematics that deals with the properties of algebraic structures called groups. These groups are defined as sets of elements with a binary operation that satisfies four axioms. Sets are fundamental in Group Theory, as they are used to define elements, subgroups, and relationships between groups. Mappings, or functions, are also important in Group Theory and are used to represent relationships between sets. Group Theory has practical applications in various fields, including physics, chemistry, computer science, and cryptography.
  • #1
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Homework Statement



Prove that f: S -> T is one-to-one if and only if f(AnB) = f(A) n f(B)
for every pair of subsets A and B of S

Homework Equations



See above

The Attempt at a Solution



Part 1:

Starting with the assumption f(AnB) = f(A) n f(B)

Let f(a) = f(b) [I'm going to try and prove that a=b for one-to-one]

f(a) is an element of f(A)
f(b) is an element of f(B)
f(a) = f (b)
So
f(a) is an element of f(A) n (B)
From assumption
f(a) and f(b) are elements of f(A) n f(B) so also elements of f (AnB)

so a, b are elements of AnB

Can't work out how to show they are equal though..

2. Assume f is one-to-one

So f(a)=f(b) and a=b

If a=b, a is an element of AnB
so f(a) is an element of f(AnB)

Likewise, if f(a)=f(b) then f(a) is an element of f(A) n f(B)

Don't know where to go from here.
 
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  • #2
For the first one, note that the assumption holds for any sets A and B. So you can choose them conveniently (hint: sets can consist of just one point).

For the second one, I suggest showing two inclusions f(A n B) c f(A) n f(B) and f(A) n f(B) c f(A n B).
One of them is trivial (if y is the image of a point x in A n B, then x is in A so it's the image of a point in A and x is in B so it's the image of a point in B, so it's definitely in f(A) and f(B)). For the other one you will need the injectivity, just write out definitions.
 
  • #3
Thanks... for the second one I'd considered looking at inclusions but couldn't work out how to implement it.

Don't understand what you mean about choosing convenient sets though, can you explain?
 
  • #4
Well, you have assumed (I hope, otherwise do it now :smile:) that f(AnB) = f(A) n f(B) for any subsets A and B of S, because that's what it said in the statement.

So to conclude that a = b, you can choose them as you like: if the assumption holds for any A and B, then in particular they hold for some A and B that you choose. For example, you are allowed to take A = B, A empty, B = S, A = S \ B, or whatever you think you need to prove that a = b.
My hint to you is: a single point x of S is also a subset {x} c S.
 
  • #5
Is for every A and B the same as any A and B?
 
  • #6
Yep, they both mean [itex]\forall A \subset S, \, \forall B \subset S[/itex].
 

1. What is Group Theory?

Group Theory is a branch of mathematics that studies algebraic structures called groups. It deals with the properties of groups, their subgroups, and the relationships between them.

2. How are groups defined in Group Theory?

In Group Theory, a group is defined as a set of elements with a binary operation that satisfies four axioms: closure, associativity, identity, and inverse. These axioms ensure that the operation on the elements is well-defined and that the group follows certain properties.

3. What is the significance of sets in Group Theory?

Sets are fundamental in Group Theory as they are used to define the elements of a group. A set is a collection of distinct objects, and in Group Theory, these objects are the elements of a group. Sets also allow us to define subgroups and study the relationships between groups.

4. How do mappings relate to Group Theory?

Mappings, also known as functions, are used in Group Theory to represent the relationship between two sets. In the context of groups, mappings can be used to define homomorphisms, isomorphisms, and automorphisms, which are important concepts in Group Theory.

5. What are some practical applications of Group Theory?

Group Theory has many applications in various fields, including physics, chemistry, computer science, and cryptography. For example, it is used to study the symmetry of molecules in chemistry and to analyze data in social networks and other systems in computer science.

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