Group Theory Sets and Mappings

  • Thread starter ZTV
  • Start date

ZTV

5
0
1. The problem statement, all variables and given/known data

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

2. Relevant equations

See above

3. 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.
 

CompuChip

Science Advisor
Homework Helper
4,282
47
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.
 

ZTV

5
0
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?
 

CompuChip

Science Advisor
Homework Helper
4,282
47
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.
 

ZTV

5
0
Is for every A and B the same as any A and B?
 

CompuChip

Science Advisor
Homework Helper
4,282
47
Yep, they both mean [itex]\forall A \subset S, \, \forall B \subset S[/itex].
 

Want to reply to this thread?

"Group Theory Sets and Mappings" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top