Let f be an application from E to E (E≠∅) such that f∘f∘f = f
Prove f is an injection ⇔ f is a surjection
I honestly have no idea how to start and I'd love to know the answer before my math exam tomorrow morning (Thinking)
1.Prove f(A⋂B) ⊂ f(A) ⋂ f(B)
2.Prove f(A) ⋂ f(B) ⊂ f(A⋂B) ⟺ f is an injection
I've solved the first question , as for the second I started with f(A) ⋂ f(B) ⊂ f(A⋂B) ⇒ f is an injection this way :
Let's suppose f(a) = f(b) = p
If we consider A = {a} and B = {b} then f(A) = f(B) = p
then f(A) ⋂...
Let A,B and C be three elements of P(E)
1. Solve in P(E) the following equation : AUX=B
2. Let's suppose that C ⊂ A ⊂ B , solve in P(E) the following system : AUX=B and A⋂X=C
I've already answered the first question , it's X = (B\A) U Y such that Y∈P(A)
As for the second , I thought maybe X=C...
Let A,B,C be three sets such that :
A={x∈ ℤ / x=11k+8 , k∈ℤ}
B={x∈ ℤ / x=4k , k∈ℤ}
C={x∈ ℤ / x=11(4k+1) -3 , k∈ℤ }
Prove A⋂B = C
I started with this :
Let x be an arbitrary element of A⋂B
then ∃(k,k')∈ ℤ² such that x=11k+8 and x=4k'
then 11k+8 = 4k'
then 11(k+1)-3 = 4k'
I don't know where...
Hello :) Thank you , I think I may get it now ?
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ⊂ (AxB) Δ (AxC)
Then I'll just try to go backwards maybe ?
Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)
I tried to start with this :
Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
But I don't know how to continue or if I should even...