When dealing with particular situations in mathematics, it is often easier to have a set which contains all the elements we deal with. This set is called a universal set and we want to define it properly.

**Definition**: Universal Set

A **universal set** $U$ is a set of elements fulfilling all the necessary or sufficient properties that we deal with in a particular situation.

In a Venn diagram, we draw the universal set as a frame in which we place the sets of our consideration (here a set $A$).

### Examples:

- If you are considering all vans, the universal set could be the set of all cars.
- The set of all possibilities to win in a coupon-based lottery could be considered as the universal set of all the possibilities printed on the coupons produced for this lottery.
- The set of all possible words which can be constructed from Latin letters is the universal set of all words which have a meaning in the Englisch language.

| | | | | Contributors: *bookofproofs* | References: [979], [7838]

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[7838] **Kohar, Richard**: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

[979] **Reinhardt F., Soeder H.**: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10

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