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I am studying discrete mathematics using Rosen Discrete Mathematics 7th Edition. I am doing sets. I don't understand what this means.

The image showing the notation

I don't understand why the intersection of all these sets is {1}. I thought it would be all {1,2,3,...., i}.

user170231
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chaulap
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    And what would you pick for $i$? –  Sep 10 '20 at 16:08
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    Hint: $$\bigcap_{i=1}^{\infty} A_i\subset A_1$$ – mwt Sep 10 '20 at 16:09
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    If something is in the intersection, it's in each set. – Sean Roberson Sep 10 '20 at 16:10
  • ${1}\cap {1,2} = {1}$. Yes? Good. Now... ${1}\cap {1,2}\cap {1,2,3} = {1}\cap {1,2,3} = {1}$. Follow? Now, generalize... ${1}\cap {1,2}\cap {1,2,3}\cap \cdots \cap {1,2,3,\dots,i}$ and further ${1}\cap {1,2}\cap \cdots \cap {1,2,3,\dots,i}\cap \cdots$ – JMoravitz Sep 10 '20 at 16:15

3 Answers3

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Remember that if $P\subset Q$, then $P \cap Q = P$. it means that in intersection the common inner parts are the result, since $P$ is all inside $Q$, then the result must be $P$. The same logic is for your example sets:

$A_1 = \{1\} \subset \underbrace{\{1,2\}}_{A_2}\subset\underbrace{\{1,2,3\}}_{A_3}\subset \cdots\subset A_n,$

So, for all $n\ge1$, we have that $A_1 = \{1\}\subseteq A_n$. Thus, $\displaystyle\bigcap_{n\in\mathbb{N}}A_n = A_1 = \{1\}$

(Just think : what elements are always inside all $A_n$ ?

L F
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Note that an element $a$ belongs to the intersection of two sets $A_i \cap A_j$ if and only if $a\in A_i$ and $a\in A_j$. The only such element given the infinite sequence of sets you're presented with is $1$ - precisely because $A_1 = \{1\}$ is the smallest (in terms of cardinality) set in the sequence, and is contained as a subset in all the others.

If you find the $\{1,2,3,\ldots\}$ for $i=1,2,3,\ldots$ notation confusing, this is just mathematial convention and the ellipsis represents an infinite continuation of elements. Note, too, that $1,2,3$ aren't always in the set, which is easy to see by plugging in $i=1$, for example.

Chubby Chef
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Think it recursively.

When $i = 1, 2, 3, \dots$ you get the sets $\{1\}, \{1,2\}, \{1,2,3\}, \dots$

Observe that $\{1\}\cap \{1,2\} = \{1\}$, then $\left(\{1\}\cap \{1,2\}\right) \cap \{1,2,3\} = \{1\} \cap \{1,2,3\} = \{1\}$. Recursively, you have the answer.

CuriousGuy
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