# Why the intersection of infinite integers is $\{1\}$?

Intersection of different sets mean that we will get only the elements that exist in each of them.

Then why intersection of all $\mathbb{Z}^+$ numbers will yield $\{1\}$?

It is clear that $1$ only exists once in the collection, and $2,3,4,\dots$ none of them match $1$, if we say we don't have duplicates.

$$\bigcap_{i=1}^\infty A_i = \{1\}.$$ $$\bigcap_{i=1}^\infty \{1,2,3,\ldots,i\} = \{1\}.$$

• Intersection is defined as an operation between sets. which sets exactly are you referring to by "all +Z numbers"? – barak manos Apr 12 '16 at 4:51
• @barakmanos ∞ (intersection of collection)i=1 {1,2,3,...,i}={ 1}. – Rafsan Mobasher Apr 12 '16 at 4:51
• I could ask what you mean by an infinite integer. The term is meaningful in some contexts, and may have different meanings in different contexts. However, I have notice some people saying things like "infinite integers" when they mean "infinitely many integers". The former is an incorrect usage. $\qquad$ – Michael Hardy Apr 12 '16 at 4:52
• I formatted some of your post. Can't really figure out what you mean by "∞ (big intersection) i=1". – barak manos Apr 12 '16 at 4:53
• @MichaelHardy Infinitely many integers. – Rafsan Mobasher Apr 12 '16 at 4:53

The only thing that is a member of all of the sets above is $1$.