Sets and functions proofs needed

[fuzuli]

Hello there,

I am extremely new to mathematical analysis and do not have an idea how to prove the following questions. Could you please give me a hand and show me a way?

Let At , t ∈ T, be a family of sets, and let X be a set. Prove the identities


http://desmond.imageshack.us/Himg62/scaled.php?server=62&filename=98278507.png&res=landing


Let A and B are sets, and let f : A → B be a function defined in A with values in B.

Is it true that f (A \ B) ⊂ f (A) \ f (B)?
Is it true that f (A \ B) ⊃ f (A) \ f (B)

 

[SteveL27]

Hello there,

I am extremely new to mathematical analysis and do not have an idea how to prove the following questions. Could you please give me a hand and show me a way?

Let At , t ∈ T, be a family of sets, and let X be a set. Prove the identities


http://desmond.imageshack.us/Himg62/scaled.php?server=62&filename=98278507.png&res=landing


Let A and B are sets, and let f : A → B be a function defined in A with values in B.

Is it true that f (A \ B) ⊂ f (A) \ f (B)?
Is it true that f (A \ B) ⊃ f (A) \ f (B)

Do you know how to prove that two sets are equal? If A and B are sets and I want to prove that A = B, I have to show two things: That A ⊂ B and B ⊂ A.

And to show each of those two things, you have to show that

a) If x is an element of A, then x is an element of B.

and

b) If x is an element of B, then x is an element of A.

Does any of this sound familiar? Can you apply it to your problems?

 

[fuzuli]

Do you know how to prove that two sets are equal? If A and B are sets and I want to prove that A = B, I have to show two things: That A ⊂ B and B ⊂ A.

And to show each of those two things, you have to show that

a) If x is an element of A, then x is an element of B.

and

b) If x is an element of B, then x is an element of A.

Does any of this sound familiar? Can you apply it to your problems?

Thank you so much for your instant reply. I think I understood your point. For example for the first one:

left to right:
if m∈ X∖⋃At => m ∈ X and m∉⋃At. so m∉At for all t∈At.
if m∈X and m∉At for all t∈At, then X\At={m} for al t∈T
then, m∈⋂(X∖At)


if m∈⋂(X∖At) => m ∈ X\At for all t∈T. then, m∈X, m∉At for all t∈At.
if m∉At for all t∈At => m∉⋃At.
if m∈X and m∉⋃At => m∈X∖⋃At

Is my notation true?

 

[HallsofIvy]

Thank you so much for your instant reply. I think I understood your point. For example for the first one:

left to right:
if m∈ X∖⋃At => m ∈ X and m∉⋃At. so m∉At for all t∈At.

Excellent!

if m∈X and m∉At for all t∈At, then X\At={m} for al t∈T

No, you can't say "X\At= {m}", you don't know if there aren't other elements in X\At. You can, of course, say "m∈ X\At for all t" and that's all you need.

then, m∈⋂(X∖At)


if m∈⋂(X∖At) => m ∈ X\At for all t∈T. then, m∈X, m∉At for all t∈At.
if m∉At for all t∈At => m∉⋃At.
if m∈X and m∉⋃At => m∈X∖⋃At

Yes, this is exactly right.

Is my notation true?

 

[fuzuli]

Do you know how to prove that two sets are equal? If A and B are sets and I want to prove that A = B, I have to show two things: That A ⊂ B and B ⊂ A.

And to show each of those two things, you have to show that

a) If x is an element of A, then x is an element of B.

and

b) If x is an element of B, then x is an element of A.

Does any of this sound familiar? Can you apply it to your problems?

and for the second one:

left to right:

if m ∈ X\⋂At => m∈X and m∉⋂At => m∈X and m∉At for all t∈T => m∈X\At for for all t∈T. Therefore m∈⋃(X\At)

right to left:

if m∈⋃(X\At) => m∈X and (∃t∈T that m∈At or ∄t∈T that m∈At)
if not for all t∈T, m∈At, then m∉⋂At. therefore m∈X\⋂At


I feel like my notations are not very good. How can I do these solutions better?