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 Question from Mac, a student: Hi, I learnt set theory recently. My teacher and few of the weblink actually give different definition for basic set. Can you please solve this ? My teacher says, {1,2,3} and {1,1,2,3} is also set. But in this link http://library.thinkquest.org/C0126820/setsubset.html it says, "A set has no duplicate elements. An element is either a member of a set or not. It cannot be in the set twice." and "{1, 2, 3} is the same as the set {1, 3, 2, 3, 1}" My question is, 1. Whether duplicates allowed in the set or not ? 2. Even if the duplicates are allowed, {1,2,3} and {1,1,2,2,3,3} are same or not ? Please help me to solve this ambiguity. Thanks mac

We have two responses for you

Hi Mac,

Your teacher and the thinkquest library are both correct. A set is a collection of objects.

Definition: Let A and B be sets. A is a subset of B, written A ⊂ B if for any x, if x ∈ A then x∈ B.

In words, A is a subset of B if every element of A is also an element of B. (Some people write rather than ⊂.)

Definition: Let A and B be sets. A is equal to B, written A = B if A ⊂ B and B ⊂ A.

Since a set is just a collection of objects {1, 2, 3} and {1, 1, 2, 2, 3, 3} are both sets but by the definition of equality they are the same set, that is {1, 2, 3} = {1, 1, 2, 2, 3, 3}. The convention is to write this set {1, 2, 3} since listing elements more than once is redundant.

I hope this helps,
Harley

A set does not have duplicate elements but a multiset does. In a multiset the number of times an element appears is important.

Penny

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