[Math] True or false. If A ∪ B = ∅ , then A = ∅ and B = ∅ .

elementary-set-theory

  1. True or false. If A ∪ B = ∅ , then A = ∅ and B = ∅ .

  2. True or false. If A ∩ B = ∅ , then A = ∅ or B = ∅ or both A and B are empty sets.

  3. True or false. (A ∪ A^c)^c = ∅ .

  4. True or false. If A ∈ B, then n(B) = n(A) + n(A^c ∩ B).

my work

1-TRUE. We see that the union of both sets presents an empty set. So this can happen only if A and B, both sets are empty sets.

2-FALSE. If the intersection of two states is an empty set, it does not mean that any of the sets is empty. It also means that there are no common elements in both A and B and these sets may not be empty.

3-TRUE. A^c means compliment of set A i.e. all elements leaving the elements of set A. So, the union of A and A^c will be U (universal set) i.e. all the elements are included.

4-TRUE. The n(A)+n(B) includes the number of elements common in both which is added twice. If we have 5 elements in set A and 4 elements in set B out of which 2 elements are common in both the sets, so, n(A)+n(B)=9

I need you confirm if my answers are correct or wrong

Best Answer

I checked the first two and I have comments:

  1. stating "this can happen only if A and B, both sets are empty sets'' is repeating the claim made in the question and is not a proof. A proof would be something like assuming that $A$ is not empty, then there is some $a\in A$ thus $a\in A\cup B$ and so $A\cup B\neq\emptyset$.

  2. If a statement is false then you should demonstrate it which a counter example , $$A=\{1\},B=\{2\}$$ would be such an example.

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