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Let $n$ be a positive integer. Describe using quantifiers:

  1. $x\in\bigcup_{k=1}^nA_k$
  2. $x\in\bigcap_{k=1}^nA_k$

My work: $i=\{1,2,3,\dots, n\}$

  1. $(\exists x),(x\in A_i)$
  2. $(\forall x),(x\in A_i)$

What I need help is explaining with words. Currently I have:

  • a) There exists $i$ for every $x\in A_i$
  • b) There always is $i$ for every $x$ in $A_i$
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    I believe your first "A" in part b of your work should be flipped over ;-) I assume you meant "for all"? – 2017-02-16

1 Answers 1

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Your answer is not correct. It should be, given $I:=\{1,2,\dots, n\}$,

  1. $\exists k\in I, x\in A_k$
  2. $\forall k\in I, x\in A_k$

In words, that is

  1. There exists a $1\leq k\leq n$ such that $x$ is in $A_k$
  2. For all $1\leq k\leq n$, $x$ is in $A_k$