What is the intersection of $\{1, 1, 2, 3\}$ and $\{1, 1, 2, 4\}$

Question

I have two sets: $\{1, 1, 2, 3\}$ and $\{1, 1, 2, 4\}$. What is the intersection of them? Is it $\{1, 1, 2\}$ or just $\{1, 2\}$?

Answer 1

As for the usual convention $\{1, 1, 2\}$ and $\{1, 2\}$ represent just the same set.

Refer also to cardinality of a set with repeating elements? and $\{1,1\}=\{1\}$, origin of this convention.

Answer 2

Note that the equality between two sets is defined as :

$$ A=B \iff A\subseteq B \text { and } B \subseteq A $$

Thus

$$\text {{1,1,2,3}}= \text {{1,2,3}} $$

$$ \text {{1,1,2,4}}= \text {{1,2,4}} $$

$$ \text {{1,1,2,3}}\cap\text {{1,1,2,4}} =$$

$$ \text {{1,2,3}} \cap \text {{1,2,4} ={1,2}} $$

Answer 3

If these are multisets rather than sets, then the usual convention would be to say their intersection was the multiset $\{1,1,2\}$

But if these are sets rather than multisets, then your question is really

I have two sets: $\{1,2,3\}$ and $\{1,2,4\}$. What is the intersection of them?

with the answer $\{1,2\}$ (which non-coincidentally is what $\{1,1,2\}$ really is as a set)