Infinite Intersection

Logic Level 4

Let there be an infinite family of sets A 1 , A 2 , A 3 , ⋯ ⊆ X A_1, A_2, A_3, \cdots \subseteq X

Let B i = ⋃ m = i ∞ A m B_i = \bigcup^{\infty}_{m=i} A_m

Which of the choices is not implied by this?

x ∉ ⋂ n = 1 ∞ B n x \notin \bigcap^{\infty}_{n=1} B_n

∃ n ∈ N s.t. ∀ m ≥ n , x ∉ A m \exists n \in \mathbb{N} \text{ s.t. } \forall m \geq n , x \notin A_m ∀ n ∈ N , ∀ m ≥ n , x ∉ A m \forall n \in \mathbb{N} , \forall m \geq n , x \notin A_m ∃ n ∈ N s.t. ∃ m ≥ n s.t. x ∉ A m \exists n \in \mathbb{N} \text{ s.t. } \exists m \geq n \text{ s.t. } x \notin A_m ∀ n ∈ N , ∃ m ≥ n s.t. x ∉ A m \forall n \in \mathbb{N} , \exists m \geq n \text{ s.t. } x \notin A_m

Agnishom Chattopadhyay by Agnishom Chattopadhyay , 22, India

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1 solution

Abhishek Sinha
Aug 11, 2016

The given condition implies x does not belong to the limsup of the collection of sets { A i } i = 1 ∞ \{A_i\}_{i=1}^{\infty} , i.e., x x does not belong to infinitely many sets from that collection.

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