Libby Owens

2022-07-15

Let n be a positive integer. Describe using quantifiers:
1. $x\in \bigcup _{k=1}^{n}{A}_{k}$
2. $x\in \bigcap _{k=1}^{n}{A}_{k}$
My work: $i=\left\{1,2,3,\dots ,n\right\}$
1. $\left(\mathrm{\exists }x\right),\left(x\in {A}_{i}\right)$
2. $\left(\mathrm{\forall }x\right),\left(x\in {A}_{i}\right)$
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 ${A}_{i}$

minotaurafe

Expert

Step 1
Your answer is not correct. It should be, given $I:=\left\{1,2,\dots ,n\right\}$,
1. $\mathrm{\exists }k\in I,x\in {A}_{k}$
2. $\mathrm{\forall }k\in I,x\in {A}_{k}$
Step 2
In words, that is
1. There exists a $1\le k\le n$ such that x is in ${A}_{k}$
2. For all $1\le k\le n$, x is in ${A}_{k}$

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