Suppose A ∩ B = ∅. (a) Consider the function f : P(A ∪ B) → P(A) × P(B) given by f(X) = (X ∩ A, X ∩ B) and the function
Posted: Thu May 12, 2022 9:22 am
Suppose A ∩ B = ∅.
(a) Consider the function f : P(A ∪ B) → P(A) × P(B) given by
f(X) = (X ∩ A, X ∩ B) and the function g : P(A) × P(B) → P(A ∪ B)
given by g(U, V ) = U ∪ V. Prove that f and g are inverses.
(b) Prove that |P(A ∪ B)| = |P(A) × P(B)|, that is they have the
same cardinality.
(c) Explain, in at most three sentences, why knowing |P(A ∪ B)|
= |P(A)× P(B)| implies there exists a bijective function h : R × R
→ R.
please solve all of them. Thank you!
(a) Consider the function f : P(A ∪ B) → P(A) × P(B) given by
f(X) = (X ∩ A, X ∩ B) and the function g : P(A) × P(B) → P(A ∪ B)
given by g(U, V ) = U ∪ V. Prove that f and g are inverses.
(b) Prove that |P(A ∪ B)| = |P(A) × P(B)|, that is they have the
same cardinality.
(c) Explain, in at most three sentences, why knowing |P(A ∪ B)|
= |P(A)× P(B)| implies there exists a bijective function h : R × R
→ R.
please solve all of them. Thank you!