Question

Let A, B, C and D be sets. Prove that A \ B and C \ D are disjoint if and only if A ∩ C ⊆ B ∪ D.

Answer #1

Let A, B, C and D be sets. Prove that A\B ⊆ C \D if and only if
A ⊆ B ∪C and A∩D ⊆ B

Let A, B, C
be sets. Prove that
(A \ B) \ C = (A \ C) \ (B
\ C).

Let A,B and C be sets, show(Prove) that (A-B)-C =
(A-C)-(B-C).

Let A and B be sets. Prove that A ⊆ B if and only if A − B =
∅.

Let A,B,C be arbitrary sets. Prove or ﬁnd a counterexample to
each of the following statements: (a) (A\B)×(C \D) = (A×C)\(B×D)
(b) A ⊆ B ⇔ A⊕B ⊆ B (c) A\(B∪C) = (A\B)∩(A\C) (d) A ⊆ (B∪C) ⇔ (A ⊆
B)∨(A ⊆ C) (e) A ⊆ (B∩C) ⇔ (A ⊆ B)∧(A ⊆ C)

Let A and B be sets. Prove that (A∪B)\(A∩B) = (A\B)∪(B\A)

Prove that for all sets A, B, C,
A ∩ (B ∩ C) = (A ∩ B) ∩ C
Prove that for all sets A, B,
A \ (A \ B) = A ∩ B.

Make a draft and demonstrate that: suppose that A, B, C are sets
such that B ∖ C and A are disjoint; suppose further that z ∈ B;
prove that if z ∈ A, then z ∈ C.

Let A, B, C be sets and let f : A → B and g : f (A) → C be
one-to-one functions. Prove that their composition g ◦ f , defined
by g ◦ f (x) = g(f (x)), is also one-to-one.

Problem 3.9. Let A and B be sets. Prove that A × ∅ = ∅ × B =
∅.
Please write your answer as clearly as possible, appreciate
it!

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