Proofs Help

[sirbillybob88]

Hello,

I have been having a bit of trouble with my proofs class and was hoping some kind soul could help with understanding these problems. The semester was fine while we were doing proofs that were mostly algebraic. With the problems becoming more and more general I am beginning to struggle.

Prove, disprove, or give a counterexample: Let F be a family of sets. Then
∪F = ∅ iff A = ∅ for all A∈F .



&

Prove, disprove, or give a counterexample: Let F be a family of sets. Then
∩F = ∅ iff A = ∅ for all A∈F .

Any help would be greatly appreciated. I have little to no idea where to begin with these.

Thank you for your time.

 

[pka]

Prove, disprove, or give a counterexample: Let F be a family of sets. Then
∪F = ∅ iff A = ∅ for all A∈F .

&

Prove, disprove, or give a counterexample: Let F be a family of sets. Then
∩F = ∅ iff A = ∅ for all A∈F .

The only way to learn to do this type of proof is to first memorize definitions.

Given that \(\displaystyle \mathfrak{F}\) is a family of sets:

1) \(\displaystyle x \in \bigcup \mathfrak{F} \) if and only if \(\displaystyle (\exists A\in\mathfrak{F})[x\in A].\)

2) \(\displaystyle x \in \bigcap \mathfrak{F} \) if and only if \(\displaystyle (\forall A\in\mathfrak{F})[x\in A].\)

What would it mean if 1) \(\displaystyle \bigcup \mathfrak{F}=\emptyset~? \)

 

[sirbillybob88]

The only way to learn to do this type of proof is to first memorize definitions.

Given that \(\displaystyle \mathfrak{F}\) is a family of sets:

1) \(\displaystyle x \in \bigcup \mathfrak{F} \) if and only if \(\displaystyle (\exists A\in\mathfrak{F})[x\in A].\)

2) \(\displaystyle x \in \bigcap \mathfrak{F} \) if and only if \(\displaystyle (\forall A\in\mathfrak{F})[x\in A].\)

What would it mean if 1) \(\displaystyle \bigcup \mathfrak{F}=\emptyset~? \)

Part of the problem is, I don't understand what the big U and upside big U mean. I knew them as union/intersection before, but when the symbol is placed in front of something I don't know what's going on. For instance A U B. I understand that fine. But, when its \(\displaystyle \bigcup \mathfrak{F}=\emptyset~? \), the confusion begins. Could you help explain that?

 

[stapel]

Part of the problem is, I don't understand what the big U and upside big U mean.

Neither your instructor nor your textbook covered the notation? Wow. :shock:

You can read up on the notation in this Wikipedia article.