Induction of Complementary Sets

  • Thread starter rbzima
  • Start date
  • #1
84
0

Main Question or Discussion Point

I'm just wondering how induction can be used to show the following:

[tex](A_1 \bigcup A_2 \bigcup A_3 \bigcup \cdots \bigcup A_n)^c = A^c_1 \bigcap A^c_2 \bigcap A^c_3 \bigcap \cdots \bigcap A^c_n[/tex]
 

Answers and Replies

  • #2
1,425
1
What's the base case? Did you try to establish whether a case follows if the previous is true (if the complement of the union of k sets is equivalent to the intersection of their respective complements, does this imply the same for k + 1 sets?).
 
  • #3
mathman
Science Advisor
7,839
439
Once you show it for two sets, it is easy. n+1 sets can be considered as 1 set union with n sets. Complement this gives then the complement of 1 set intersected with the complement of the union of n sets.
 

Related Threads on Induction of Complementary Sets

  • Last Post
Replies
9
Views
3K
Replies
5
Views
1K
  • Last Post
Replies
6
Views
699
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
9K
  • Last Post
Replies
2
Views
975
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
9
Views
4K
  • Last Post
Replies
1
Views
1K
Top