Question

请教两道离散数学问题

证明
1.Let A, B, and C be sets. Prove that
A∪包含 (A∪B ∪C).
2. Let A, B, and C be sets. Prove that
(A-C)∩(C -B) = 空集

Answer 1

1、
Pick a∈A∪B ,then a a∈A or a∈B.
there are two cases:
case 1 : a∈A, then a must be a member of
one of A,B,C. that means a a∈A∪B ∪C
case 2: a∈B, similarly discuss.

so in both cases, a must be member of A∪B ∪C
that means A∪B is subset of ∪B ∪C

2.Pick a∈(A-C)，a must be in A and not in C. because a is not C, a is not in C-B.

so, for every element a , a can not be in both (A-C) and (B-C). that means (A-C)∩(C -B) has no element.

therefore,(A-C)∩(C -B) is empty set.

Answer 2

1，如果证明A∪B包含 (A∪B ∪C)
任取x属于A∪B
它要么在A中,要么在B中
所以也一定在 (A∪B ∪C)中
所以成立

2，因为(A-C)里的元素是不属于C的，而(C -B) 中的元素都是属于C的
所以它们不相交，所以是空集