The union of a subset and its complement

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In summary, the union of a subset and its complement is the set that contains all the elements that are in either the subset or its complement. This is denoted by the symbol "∪" and is read as "union". The original set contains all the elements that are in both the subset and its complement, while the union of a subset and its complement only contains the elements that are in either the subset or its complement. This union can be empty if the subset and its complement have no elements in common, and it can be used in various set operations such as complement, intersection, finding differences, and testing for equality. In mathematics, the union of a subset and its complement is an important concept in set theory and has practical applications
  • #1
chipotleaway
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If S is a subset of X, and cS is the complement of S with respect to X, is the union of S and cS equal to X? Seems like a no-brainer but just want to be sure because I've yet to find a book that comments on this.
 
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  • #2
Yes. You can pick any element of X and prove that it is in the union of S and cS. Use the definition of "complement" and "union"
 
  • #3
Haha appropriate username, thanks. How come this isn't mentioned in most books? Seems like all the 'obvious' stuff is.
 

FAQ: The union of a subset and its complement

1. What is the union of a subset and its complement?

The union of a subset and its complement is the set that contains all the elements that are in either the subset or its complement. It is denoted by the symbol "∪" and is read as "union".

2. How is the union of a subset and its complement different from the original set?

The original set contains all the elements that are in both the subset and its complement, while the union of a subset and its complement only contains the elements that are in either the subset or its complement.

3. Can the union of a subset and its complement be empty?

Yes, if the subset and its complement have no elements in common, then the union of the two sets will be empty. This can happen when the subset is the empty set or when the subset and its complement are mutually exclusive.

4. How can the union of a subset and its complement be used in set operations?

The union of a subset and its complement can be used in set operations such as complement and intersection. It can also be used to find the difference between two sets or to determine if two sets are equal.

5. What is the significance of the union of a subset and its complement in mathematics?

The union of a subset and its complement is an important concept in set theory. It helps us to understand the relationship between sets and their subsets, and how they can be combined to form new sets. It also has practical applications in various fields such as statistics, computer science, and engineering.

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